Kyoji from Akanebanashi invokes this. He's a Rakugo-kanote a storyteller who conveys (usually) comical tales through pure verbal performance but has a naturally serious personality that is quite the obstacle in a field that requires him to entertain his audience. His teacher advices him not to try to force himself to crack jokes or be witty, but instead to play up his seriousness to ridiculous extremes that it becomes hilarious.
Attack on Titan has Captain Levi, who doesn't crack a smile ever. He's also the best at what he does (killing Titans) despite being one of the shortest male characters in the series. He's also deadly serious about cleaning, wears an apron and a dust mask and it just looks hilarious.
Mikasa Ackerman also gives a lot of this. Most funny moments involving her revolve around the fact that shes known to hold an absurdly serious and deadpan expression in (almost) any situation, along with the occasional dry quip. The "Sasha farted" in boot camp currently provides the page image, and then there's when her squadmates ask her exactly happened when Eren lost control during his second Titan-shift and attacked her, punching in his own face in the process, she claims he was swatting a fly. And that's nothing compared to her Clingy Jealous Girl moments (which are ramped up in official parodies and the Attack on Titan: Junior High spinoff).
Azumanga Daioh: Sakaki is notable for her perpetual stone faced expression and monotone. Yet, she delivers some of the most bizarre lines and has equally weird daydreams, like imaging Chiyo's dad as an oblong talking yellow cat.
This is discussed in Bakuman。, where after one too many failures in getting serialized, Ashirogi Muto's then-former editor Akira Hattori gives them a tip to help set them in the right direction: serious humor. Later on, Takagi decides to bring Mashiro with him to trail Mr. Hattori for a whole day, and at the end, they realize what "serious humor" means after witnessing a few examples along the way (such as Iwase coming on to him and him rebuking her, both keeping themselves straight throughout), which leads to the creation of Perfect Crime Party, the series that gets them out of their creative rut.
Black Clover: With the exception of conversations with Asta, everything about Yuno is deadpan. Everything. Even Sylph tugging on his cheeks. The effect is frequently hilarious.
In the omakes, the first time Byakuya makes a joke, Renji freaks out. Ignoring Renji's advice, Byakuya decides to do a comedy routine on stage. As Renji expected, the audience is deeply disturbed. His artwork is as bad as Rukia's, but whereas Rukia is obsessed with bunnies, Byakuya is obsessed with something of his own creation - a childishly drawn... thing he calls Ambassador Seaweed. The Ambassador Seaweed freaks everyone out.
When Squad Zero drops into the Soul Society after its surprise attack by the Vandenreich, Senjumaru is playing several instruments with her extra arms to herald their arrival, while she herself has a completely deadpan expression.
Buddy Daddies: Rei doesn't waste any energy on expressing himself, so he only ever shows a resting bitch face and uses a flat, hushed voice. Dressing up as a lama mascot, playing with Miri on a seesaw or Kazuki dragging him away by the collar barely elicits a response. He is well aware of how perfect it is for snarking at Kazuki's Drama Queen moments.
Code Geass: Part of the appeal of Lady of War Cornelia li Brittania is the potential for comic seriousness in just about any comedic situation she could be put into. Fanartists have caught onto this, and she's frequently shown looking flustered in comparison to the other characters. Canon toys a bit with this, considering the reaction she had to Clovis's raunchy swimsuits. And Cornelia was naked or at least topless when she commented on how Euphie shouldn't wear any of them. She was concerned about wearing one herself, and it was quite amusing to hear Euphie forcibly putting one on her.
Lelouch has his moments too. The Dark Messiah is capable of screaming like a little girl when a cat steals his mask, before running after the cat proclaiming "I shall not be defeated by a feline!" as passionately and seriously as he often declares "I will obliterate Britannia!" As with Cornelia, the sound dramas have fun with this. Suzaku in particular enjoys teasing Lelouch in order to get him flustered. Lelouch goes from panicking about the potential downfall of the Black Knights to blushing because Suzaku ate a piece of rice off Lelouch's face within the space of a minute.
Origami in Date Alive refuses to smile, laugh, cry or anything no matter what situation she's in, resulting in a lot of this. For example, in the first OVA, she barely responds to Shido claiming to be a mom-con/sis-con/pedo (he was attempting, and miserably failing to make her fall out of love with him) and later, with a completely straight face, dresses up in a school swimsuit and dog ears and tail then puts Shido's belt around her neck like a collar after he orders her to. Shido, the crew of the Fraxinus, and the crowd are completely freaked out, but Origami just doesn't care.
Death Note: L can sometimes lapse into this, such as a random moment when he gravely tells a detective, "I'm going to give you this strawberry. Please keep it secret." The detective dutifully slips the strawberry into his pocket.
Don't Toy with Me, Miss Nagatoro: Sana Sunomiya, aka the Arts Club President, as she treats everything seriously while everyone else is flustered, intimidated or otherwise taking it more lightly, especially when it comes to art. Her cousin Hana takes over the role after she graduates, having very similar wacky perceptions about art.
Re-L in Ergo Proxy gets this a couple of times, due to her being a Very Serious Intelligence Operative... who ended up Walking the Earth with Vincent, who is effectively The Fool with superpowers.
Hayato Akaba of the Bando Spiders remains almost completely stone-faced while striking goofy rock-star poses for no apparent reason, and has a tendency to make convoluted music metaphors with the same completely serious expression.
Shun Kakei of the Kyoshin Poseidons always takes the antics of his wackier teammates in stride, whether it's the rivalry between Ohira and Onishi or whatever hijinks Mizumachi is up to.
Erza Scarlett from Fairy Tail, who reacts with deadpan seriousness to almost all the wackiness the guild gets into. It's particularly telling (and amusing) when she's around goofballs like Natsu and Happy, who are among her closest friends, as she barely bats an eye at their antics and generally takes them at face value. This is in contrast with Lucy and Gray, who are only too quick to call them idiots. Really the only time Erza drops her natural stoicism is when she's around her Love InterestJellal.
Somehow, Unfunny Hatori Sohma from Fruits Basket manages to put up with Chivalrous Pervert Shigure Sohma and Large Ham Ayame Sohma. Hatori is the only serious one of the three, without much of a sense of humour. Although this is justified, given his traumatic past. Maybe he's just plain tired of Shigure and Ayame's antics, after all this time?
Riza Hawkeye is a rare female example in the manga omake, doing such things as trying to measure Ed's height against his will, introducing her dog and responding to Mustang's musing that he might want to grow some facial hair by drawing on his face with a marker all with a completely straight face.
In Fullmetal Alchemist (2003), Roy Mustang is like this up until he's confronted with Black Hayate and gives a speech about how much he loves dogs! Later in the episode, he reveals that his first executive order as Fuhrer President would be to make all female army personnel wear... tiny miniskirts!
Full Metal Panic!: Sousuke Sagara. At least it tends to actually be his fault, though. He eventually plays the straight man to his own mecha.
Juubei Kakei in Get Backers (at least as an adult). Toss in blindness, unfamiliarity with the modern world, and copious amounts of Ho Yay. And for extra hilarity, they told him he wasn't funny. He was devastated.
The Shogun, Shigeshige Tokugawa, is an Iron Butt-Monkey who ends up naked, accidentally beaten, or in particularly ridiculous situations every time he appears, but what makes it even funnier is that he always keeps a perfectly stern face through all of this.
Katsura spends most of his time making a fool of himself by doing things that are considered crazy even by the other characters of the show, coming up with absurd plans or appearing in public in very suspicious costumes. But the joke with him is that he takes everything he or anyone else is doing extremely seriously, no matter how stupid or insane it might be to anyone with more common sense.
All the girls of Kuromorimine from Girls und Panzer are shown to be this. Best exemplified in the Motto Love Love Sakusen Desu manga where Miho invites them to a Christmas party at her house but they assume it's a performance review and panic until she arrives. This is not helped by the fact that Maho is a stoic Ice Queen who has no idea how to organize a Christmas party because it was usually her younger sister, Miho, that handled it.
The title character of Goblin Slayer. In a blood and violence soaked series full of rape Goblins, 90% of the rare humor comes from Goblin Slayer acting like a borderline automaton. Especially in regards to how food just passes through the slots in his helmet.
Raven from Gravion, though this is a result of literally Becoming the Mask. Ayaka, the current "Raven", is a woman and far more emotional than her unfunny alter ego.
Kyon, who is also the Deadpan Snarker; though his creativity and sense of wonder are probably just repressed. His aloofness doesn't prevent him from getting stuck in alternate worlds with a MagicalShe Is Not My Girlfriend, or from finding himself smack-dab in the middle of mysterious murder cases.
Yuki Nagato is another example, reacting with robotic coldness to all the wacky hijinks created by Haruhi and Mikuru. Yuki herself is not above a friendly prank or two. Even if she doesn't show external signs of enjoying it...
Kyon's genderbent counterpart, Kyonko, is even better at this than the original.
Germany, of course. How could we forget about him? He is most likely the biggest example of this trope in the series.
Switzerland is generally a serious, stern character, and occasionally throws out gems like this in his typically serious, stern manner:
Switzerland: Christmas is about family and loving people. [Pause.] Now get bent.
Japan has his moments as well, especially during Germany's training sessions on how to deal with being asked to react if England attacks, or what to do if the enemy asks him to surrender. He gives his answers in complete seriousness:
Japan: You should imitate William the First's loud voice! Japan: Yes sir! I respond Japanese way! Be unclear! Say one thing, but mean something else completely opposite! Lie to them!
Laura from Infinite Stratos takes everything seriously. Whether it be an IS battle or trying to find a pair of "hot panties" to flash Ichika with, she will treat it all the same. Bonus for her being German.
In the Clear Moonlit Dusk: In addition to her chaste and princely behavior, Yoi is in general a serious, straight-forward, and somewhat dense person, leading her to react to even the lightest of ribbings and teasings with bewilderment and getting flustered. For example, she takes Ichimura's suggestion to test date as Serious Business, producing a Comically Wordy Contract at school to go over how the arrangement should be. One of Ichimura's friends asks Yoi's friend Nobara if she's always like this, to which she replies "Well, relatively."
JoJo's Bizarre Adventure: Jotaro Kujo has a perfect poker face, as he demonstrates to good effect when gambling against D'Arby. The reason he's so stoic is that he thinks his emotions are easy to read, so he tries to hide them, but of course no one can tell what he's thinking anyway. Naturally, he's put into quite a few humorous situations thanks to his deadpan reactions to the bizarre things that happen in the series.
Lupin III has the very formal and traditional Goemon Ishikawa XIII, who is often used this way for the Bathos. In the crossover movie, Lupin III vs. Detective Conan, Goemon jumps from a plane and cuts through several floors of a building on his way to land in the vault. Once there, he says "Once again I have cut a worthless object" as usual, completely serious... and his legs are wobbling from the impact, seemingly unnoticed. He tries so very hard to be serious and dignified... but when he hangs out with Lupin...
Lyrical Nanoha: Chrono's expression of cold professionalism remains the same even when he's wearing a bandage tied as a ribbon, combing down a coworker's Idiot Hair, or lightening up the atmosphere with a joke. He grows out of it by the time that he's an adult.
Mash in Mashle: Magic and Muscles is a Deadpan Snarker who takes what other people say way too literally, comments on how absurd things are without emoting, or just outright ignores people.
Shoto Todoroki will occasionally say or do seemingly absurd things with complete seriousness and sincerity. After the Stain arc he muses about whether, between Midoriya and Iida, he had some curse that causes people to maim their hands around him. It takes a moment for the other two to realize he's serious. Later, when everyone sees he apparently rebuilt his dorm room with tatami mats in a single day, his only response is he worked hard. Indeed, he's noticeably more tired than the others.
When doing supplemental training for his provisional hero license, Todoroki is one of three students accosted by a teacher in full Drill Sergeant Nasty mode. He calls them all pieces of crap and demands to know how they could ever expect to help people. Todoroki actually considers the question and begins giving an answer about use as fertilizer before being smacked down for needing more guidance.
His father, Endeavor, is this as well albeit in a different way. He's not as Literal-Minded as his son, but he does tend to comically overreact to certain situations, particularly in his dynamic with #2 hero Hawks. Also, because of his Good Is Not Nice attitude, he tends to freak people out when he tries to genuinely act nice and goes completely Large Ham when it comes to dealing with things involving Todoroki.
Fumikage Tokoyami uses Sesquipedalian Loquaciousness to refer to otherwise mundane things (like calling a recreational game at their forest training camp "a mad banquet of darkness") and takes great pride in his Chuunibyou behavior; most notably, when his goth-themed room wasn't taken as seriously as he expected, he was quite angry.
Tenya Iida is somewhat more lighthearted than Todoroki, but he still takes most things very seriously, resulting in plenty of comedic moments, such as his reactions to his classmates goofing off, or Midoriya getting punched in the balls by Kota at the training camp. His status as the class representative only amplifies this.
All Might's former sidekick, Sir Nighteye, places high value on humor but you wouldn't know it to look at him. He's a stern man whose hero costume is literally just a suit and tie. He will, without a change in expression, put his own sidekicks in a tickle torture machine if he thinks they aren't being funny enough and uses seals, stamps used in Japan in place of signatures, as his preferred weapon, noting the humor of how it goes along with his "salaryman" look.
Sasuke Uchiha and Neji Hyuuga work out like this, though moreso in the anime. Of course, when something funny actually happens to them, it's just that much better. Seeing Sasuke with those white circle eyes is good for a laugh.
Best example of this has to be Shippuden episode 181 (a filler) where Sasuke gets attacked by an ostrich.
One of the few fillers generally considered worth watching has Neji spazzing out when he eats curry. Were it anyone else making those expression, it wouldn't be as funny. This trope is probably why Episode 101 is so popular despite being filler.
Also, Shino. Is it really anyone wonder they dedicated an entire filler episode to getting drugging him to laugh? His return of him sulking because Naruto failed to recognize him right away. Also his reaction to his bugs being eaten by that anteater.
And Kakashi. Remember his reaction when Lee takes off his heavy wrist/leg bands upon Guy's request? As everyone is all OMG WTF BBQ, he just covers his masked face and thinks "Guy, you are too much..." To enjoy a true "what the?" reaction from Kakashi, see the beginning of his fight against Naruto and Sakura in Shippuuden.
And Sai, too. His reaction to Konohamaru's yaoi version of the Sexy no Jutsu, which involves him and Sasuke naked? "Ah, it's me and Sasuke-kun."
Itachi. His response to Naruto telling him about how inescapable, completely fatal the attack being used on them is, based on his own experience? "So, why are you still alive?" The Sunny-Side Battle! OVA takes this a step further by showing him chasing a chicken... while wearing his Akatsuki robes!
Occasionally Setsuna acts like this. Her mentor-figure Eishun played this to comical absurdity in his own group.
Fate, at one point, gets his arm cut off. He's not worried at all, since apparently, he can regenerate/reattach it perfectly well, but his haremettes promptly freak the eff out. How does he try to calm them down? Use his severed arm to pat the head of the Cat Girl of the harem. All with a rather dead expression on his face. It's rather hilarious. Two of the other haremettes actually point this out:
Shirabe: F... Fate-sama actually attempting physical humor? Homura: Is that supposed to be funny?
Chisame also qualifies on some occasions; most of her humor concerns her reactions to whatever absurd situation she ends up in.
And in Cosmo Entelechia, we have Dynamis, who can be a really really Large Ham, but is always dead serious.
Gendo can also stare depressingly at Eva-01... while in an orange raincoat.
There's also something funny about Fuyutsuki facepalming and muttering "Those kids are embarrassing us again..." when Shinji and Asuka are bickering at the end of Episode 9.
In the dub he says "Those darn kids are embarrassing us again". There's something very comical about that, as the wizened old Vice-Commander can't even say "damn". It comes out sounding like a line from a sitcom.
Rei Ayanami, several times, specially whenever she interacts with Shinji and/or Asuka.
Zoro. He plays both "badass" and "buffoon" very well, but what makes it funny is that he's so serious the whole time. One example is when he, Nami and Vivi are caught by Mr. 3 during the Alabasta arc and placed in a Death Trap with powdered wax pouring over them until they become living wax statues. The girls are in despair, but Zoro keeps his cool throughout. Eventually, he decides to raise his sword and point it to the sky, justifying himself with something along the lines of, "if I'm gonna die like a statue, might as well make it look cool". All the while keeping his straight face.
A filler episode set soon after Enies Lobby uses this as its entire premise (the title, "Zoro's Slapstick Housework Help", should tell you everything you need to know)
He borders on being a Comedic Sociopath. He's a cold-blooded killer, but when he angrily shouts at and threatens someone for giving him directions when he gets lost on a straight, narrow cliff, you know you're supposed to laugh at him.
Zoro and Usopp are cuffed together while facing two opponents. With a totally serious face throughout. Zoro still is completely serious even as he executes his (innovative but completely ridiculous) plan. See it to believe it.
Zoro: I have a plan. We play rock-paper-scissors. The loser cuts off his hand. Usopp: How can you say such a scary thing with a straight face! Zoro: I'm not finished. Then he goes and gets Chopper to sew it back on. Usopp: We're not rag dolls!! Zoro: I have another idea... Usopp: I don't wanna hear it!!!
After the timeskip, we see he hasn't lost his touch. In Sabaody, when right after he cuts a whole galleon ship in half and the crew berate about it, he calls it "a quirk of fate" and then says in a completely deadpan tone "I got on... the wrong ship". Then while on the way to Fishman Island, his reaction to seeing the Coribou Pirates floating away after the Kraken destroys their ship is to say "They look like jellyfish".
Nico Robin is another example, especially given she's the only crew member who never adopts crazy facial expressions when shocked or angry (though that's been eroding since the Enies Lobby arc, and she's actually had a genuinely shocked face in the Dressrosa arc). She once calmly remarked how she knew Zoro hadn't been eaten by a shark because the water hadn't turned red, much to her crewmates' shock. She's also an occasional Straight Man for the crew's antics.
And a less prominent character who is a notably perfect example: During the Enies Lobby showdown, Cipher Pol 9 agent Kaku takes this trope to hilarious lengths (all ye who have not watched his fight with Zoro, spoil at your own risk): not only does he turn into a ridiculous block-nosed giraffe-man, he gets defensive about it, and says things like "Witness the power of a giraffe!" all while keeping a perfectly straight face. Using attacks like "Giraffe Canon" and "Nose Gun" is several times funnier just because of how absurdly seriously Kaku takes himself. No wonder he came in 9th in the following popularity poll. Though he's right about being dangerous.
Jinbei is starting to evolve into one by virtue of hanging out with Luffy and Hancock on Amazon Lily.
Trafalgar Law does not appear to be sure how to react to the Straw Hats' usual antics when he offers to ally with them in Punk Hazard, resulting in this trope.
Pica of the Donquixote Pirates. Of all the members from said pirate crew, he is The Stoic. However, despite being such a serious character with no quirks, he is a huge, Top-Heavy Guy with a very high-pitched voice. Hilarity Ensues whenever anyone hears his voice and starts laughing, much to his annoyance.
Out of Sanji's Big, Screwed-Up Family, the Vinsmoke clan, the one who fits the most is the eldest son Vinsmoke Ichiji. Seeing the aloof and snarky Warrior Princegoing gaga over Nami like his estranged brother Sanji did after seeing her for the first time was pretty funny. Seeing him with big heart eyes while keeping his serious disposition is even funnier.
Both Zoro and Law continue this attitude in the Wano arc. The former learns that all the animals and water are poison, and simply responds "No wonder my stomach has been hurting", and the latter shows up to fight Basil Hawkins dressed as a Komusō.
One-Punch Man: Most of the humor involving Genos is that he's dark and serious about everything (like paying rent and peeling potatoes) in contrast to Saitama's apathy and lightheartedness.
Pani Poni Dash!: Rei Tachibana. Nothing makes her humorless nature more glaring than when her pissed-off face takes up two thirds of the screen while Himeko rises up on the remaining one-third and shoots off her motormouth while rocking side-to-side.
Tsurugi Inugami would count even more, with him being the serious, near-emotionless one in the same class as characters like Misao Nanjo and Behoimi (not to mention that said class is taught by Old Geezer).
Ichijou never once changes facial expressions (except to change to a more super-deformed version of the same stoic expression,) but is the Cloudcuckoolander supreme of the series.
Panty & Stocking with Garterbelt: Garterbelt, Panty and Stocking's stern taskmaster. Ghost-slaying at God's command is Serious Business, and there's nothing absurd about ghosts of turd, semen, boogers or vomit. There are a few moments when even he can't remain serious all the time though. The end of the episode "If Angels Wore Swimsuits" is one of the funniest scenes in the whole series because of him.
Giovanni from Pokémon: The Series. Which of course, only makes Meowth's progressively delusional Imagine Spots that much weirder.
The Team Rocket trio themselves in the Black & White seasons. One episode had them break from their serious dispositions when their life forces were being drained by Litwick. They attempt to menace the twerps, but they are exhausted and gasping for air even while reciting their motto.
The entirety of Prison School can be summed up as this. The realistic and serious art style is juxtaposed with the absurdity of the premise and plot. The Chairman in particular is always stone-faced and over-dramatic... While also being a porn-addict who has a bad habit of having his Porn Stash exposed.
Queen Millennia: Yayoi replies to Hajime that the hunt for the meteorites would cause a "hullabaloo" if they weren't important without changing her expression. Later she tells Hajime she's "dead serious" about being Queen Millenia.
Rooster Fighter: By just being a rooster, Keiji turns every scene he appears in into a comedy by treating every situation with intensity, sometimes even bewildering his companions.
Sailor Moon: Mamoru Chiba/Tuxedo Mask is a pretty serious guy, but can sometimes lapse into this. Particularly when his child-like girlfriend Usagi/Moon (or any of the Sailor Senshi) say or do something that is so over-the-top all he can do is stare at them with a look that says "Moving right along..." Haruka/Uranus and Michiru/Neptune also slip once or twice into this.
Jin in Samurai Champloo. At first glance, he comes off as a stoic, cool, smooth, handsome badass, but then the series proceeds to place him in some of the most comedic of situations alongside his companions. This includes a torture montage, working at an eel stand, getting a sore back after a night with unattractive courtesans, having to pretend to be a tour guide to win back his pawned swords, and dressing up as a female prostitute.
It's the fact that he's always so damn serious that gives him the most hilarity of this sort in the entire flippin' show. He's kneeling on a rock in the middle of a river while Mugen and Fuu are over on the shore. With no warning he suddenly dives in, swims and flails around like an utter madman, RUNS up to them in double-time animation (freaking them out in the process) and collapses in a prone-kneeling position before them, panting heavily... And holding a pouch full of (unknowingly, forged) gold coins. MUST be seen to be believed.
Science Fell in Love, So I Tried to Prove it: Probably the most common gag of the series is to have Yukimura and/or Himuro begin ranting about the science and statistics behind minor things while delivering every word as if they're giving lectures on nuclear physics, coming off as complete Cloudcuckoolanders to almost everyone else around them.
Most of the punchlines from Servant × Service's Megumi Chihaya come at her outwardly calmer moments... such as declaring dryly that she and Ichimiya are an Official Couple.
Shimoneta: Oboro is head of the Decency Prefects, which makes it "her" job to crackdown on SOX's ero terrorist activity... by confiscating the school's basketball hoop and volleyball net for even vaguely resembling a condom and fishnet stockings. Yes, you read that right.
Slayers: Zelgadis shifts between this and being the group's Only Sane Man. Which naturally makes him the whipping boy whenever Lina needs a diversion, or a scapegoat to vent her frustrations on.
Stein from Soul Eater with his Running Gag of him not be able to slide in a office chair without falling.
Let me try that again.
While Stein does have his moments, Medusa plays a much straighter example. Everyone in the world of Soul Eater has a comedic gag going for them. Medusa's is interacting with these people while being the most serious person in the series. A key example is her meeting with Death and Spirit.
SPY×FAMILY: Twilight, AKA Loid Forger, has the ability to drop into spy mode to seriously analyze every possible facet of whatever situation he's in, even if it's something extremely mundane (trying to convince his daughter to study) or extremely ridiculous (Anya trying to convince him to try on every available outfit in a store at the same time).
Superman Vs. Meshi: Batman, who remains stoic and frowning as Superman poses horizontally so he can fly him to Japan. He opens up to Superman once he tastes the restauarant's delicious food.
Tono to Issho: Most of the main characters just don't realize how insane the stuff coming out of their mouths sounds, and remain completely stone-faced.
The Way of the Househusband has its lead character "Immortal" Tatsu, an ex-yakuza turned House Husband now dealing with ordinary household tasks like shopping or cleaning without having changed any of his mannerisms.
Zoro: I have a plan. We play rock-paper-scissors. The loser cuts off his hand.