For obvious reasons, all examples contain spoilers, so they will be unmarked.
General:
Fanfics in general can include late-arrival spoilers from the original works. If someone new to a fandom starts looking, they may be surprised by summaries that give away important plot points.
A sequel to any fanfic can also have this effect if the author includes spoilers in the summary.
Specific works:
Blackbird (Arrow): By Chapter Three, most of the readers have guessed that the "terrible choice" Dinah made was trading Laurel to the League of Assassins in exchange for Sara's freedom.
The title of the second installment in the Blueblood series, Blueblood's Redemption, implies Blueblood wasn't redeemed in the first, Blueblood Returns, essentially spoiling its ending that his Heel–Face Turn wasn't genuine. Unfortunately, anyone looking up the fic now is likely to find it listed along with the sequel.
Brendans Pokemon Adventure has spoilers for the next few chapters, and has a page exclusively for spoilers.
In Blizzard of the Red Castle, Akagi being Fubuki's mother, and Fubuki's ignorance of that, is the core source of drama, and it's revealed to the audience at the end of the first chapter.
Dungeon Keeper Ami assumes the reader has finished the first Dungeon Keeper and at least played the sequel, as many events from the story take place in the same scenarios where the first game was played and the differences in gameplay from the two games become an integral part of the plot.
It also assumes the reader has at least finished the first arc of Sailor Moon for the sake of some characters; like Jadeite being locked into Eternal Sleep by Queen Beryl.
Very early on in Eddward Wright: Ace Attorney, minor spoilers to the ending of Port-Ed 2 exist, such as the Eds still having their ASHPDs and Edd having his Companion Cube. The latter plot point may not be as surprising for those who finished Portal 2, but the first one is. Port-Ed 2 wasn't even halfway finished when Eddward Wright started.
Equestria's First Human: The askblog based around the fic, as well as the author's DeviantArt page, make it pretty obvious that Connor becomes a pony.
Evangelion 303: If someone checks Grummancat's page before reading all chapters in order, that person will get spoiled that: Jessika was in love with Asuka, Unit-04 crashes and Jessika dies, Asuka falls in a coma but it only lasts several months, Asuka's suicide attempt fails, Shinji and Asuka got engaged and Mari Makinami is the Sixth Child.
Facing the Future Series: Anyone who reads ahead in the series will learn that Sam has become half-ghost and that Danielle has been adopted by the Fentons.
The summary and first couple of chapters of The Legacy of the Fire Empire (the sequel to The Fall of the Fire Empire) spoil both the ending of the first story, as well as the major plot point of Aang still being alive, after waking from the coma-like state that the Fire Empire's actions kept him in.
Hinterlands centers around a group of bounty hunters trying to catch a necromancer named Amanita. The very premise of its sequel, Urban Wilds, spoils that Amanita is The Atoner with a case of Bad Powers, Good Ponies.
Infinity Train: Boiling Point takes place after "Wing It Like Witches" so it doesn't bother to keep Amity's crush on Luz a secret. And like the fic below, it doesn't keep the Apex's existence a secret.
Infinity Train: Blossoming Trail spoils a few things in regards to how Ash is now the Alola League Champion (as it takes place during Pokémon Journeys) but its prequel, Infinity Train: Knight of the Orange Lily spoils many of the twists of the Nihilego arc (namely that it's Faba's fault for the entire mess) and that it starts on the night before Professor Kukui and Burnet's wedding. And on the Infinity Train side of things, the events of Book 1 and 2 are spoiled, alongside Book 3 since it takes place at the same time.
Karma in Retrograde assumes the reader is caught up with the manga, giving away details like All Might's retirement and Overhaul's defeat.
Life in Manehattan: It's revealed about halfway through the first story in the series that museum curator Honey Do is actually the real life Daring Do, whose exploits are the inspiration for the popular book series, but she keeps this a secret so that no one will treat her like her brash fictional counterpart.
My Brave Pony: Starfleet Magic has a rather infamous moment when Twilight Sparkle, the usual protagonist of My Little Pony: Friendship Is Magic, dies to one of Season 3's main villains. And unlike most examples of this, the author makes no attempt to hide it from first-time viewers; blogs and comments of his casually mention her death as if it's something to be proud of. On that note, if you're reading Fall of Starfleet, Rebirth of Friendship or My Brave Pony: Starfleet Nemesis, just remember that the above event is canon in those universes as well. Two original characters in the former have a striking resemblance to Twilight, and a main character in the latter is a clone.
Season 2 of The New Adventures of Invader Zim assumes that readers are already familiar with the fact that Zim is organizing a rebellion against the Tallest, a plot point which doesn't come up until near the end of Season 1.
The second work in the series, Ruby Stars, spoils a twist from the first work in its synopsis, and features its repercussions as a major plot point.
Later chapters in Ruby Stars casually incorporate the original show's reveal of Rose Quartz being Pink Diamond, which took years and a curveball before it happened.
Us and Them: Reading Life Renewed, Love Renewed or any of the side stories will spoil Aeris and Sephiroth's fates.
A Voice Among the Strangers reveals towards the end that Ebony has been the new Changeling queen for some time now, and is also named Chrysalis. The first chapter of the P.O.V. SequelA Stranger Among the Voices is from Ebony's point of view and begins some time before the first story. As such, it does not attempt to keep the secret.
Webwork: A major plot point is Jade's further transformation from a Shadowkhan into a Jorogumo. This happens just late enough to spoil things for new readers, but early enough to essentially drive the whole plot.
While not nearly as egregious as some examples, in Winter War, Byakuya's survival is one of these. The series index lists the chapters by name and POV character. Byakuya is MIA at the start of the fic, his death apparently confirmed several chapters later... and then we get a chapter titled "Byakuya: Necessity." So if you started reading late, and looked at the chapter index, you probably knew he wasn't really dead even before you got to the point where Hanatarou remembers seeing him apparently die.
The first half of season three of Yu-Gi-Oh! The Abridged Series is called The Cancelled Series, which is advertised as such and, therefore, spoils what happens at the end of season two.
Pokémon Reset Bloodlines: Due to the amount of Expanded Universe sidestories, several of these follow certain plotlines and might spoil events that happened in others previously written. Examples include:
The Ghosts of Maiden's Peak Interlude already takes into account that Agatha's Gengar is actually her younger brother Tony, who was killed and had his soul turned into a Gastly during the events of the Agatha & Sam Gaiden.
Butterfree's two interludes as well. The first one has him go in search for the Pink Butterfree and finding out that she found herself another mate, while the second one deals with him still dwelling on the aftermath of that.
Voyages of the Wild Sea Horse: Discussion of anything relating to the story after the events of the East Blue arc gives away any number of things that spoiler newcomers:
Shampoo and Nabiki eat Zoan-type Devil fruits.
Ryoga swaps out his Jusenkyo Spring of Drowned Pig curse for a Spring of Drowned Girl curse.
Ranma agrees to openly try and see Shampoo, Ukyo, Kodachi and Nabiki as girlfriends. Nabiki even decides to seriously vie for Ranma's hand.
exist, such as the Eds still having their ASHPDs and Edd having his Companion Cube. The latter plot point may not be as surprising for those who finished Portal 2, but the first one is. Port-Ed 2 wasn't even halfway finished when Eddward Wright started.