While most of the play has a serious tone, the musical number in which Sweeney and Mrs. Lovett start harvesting and serving human flesh is extremely silly. It's a flurry of corny jokes about how a man's occupation might relate to his flavor, some incredibly corny puns, and a rhyming contest.
The canon of Joe Orton; every single character he's created is an amoral monster who will either kill you or fornicate with you regardless of gender.
And considering that in Real Life Joe Orton was killed by his gay lover, it seems his work was truer to life than he would have liked...
Christopher Durang specializes in this kind of humor. Two of his better known plays are The Marriage of Bette and Boo and Sister Mary Ignatius Explains It All For You. In the first, one of the title characters repeatedly bears stillborn children; the doctor, announcing their births, drops them on the floor. The second ends with the eponymous nun shooting two people.
The Cripple of Inishmaan by Martin McDonagh tells the story of a poor, crippled orphan who tries to break into show business. Everythingthat could possibly go wrong for him does. It's simultaneously one of the most depressing plays you will ever read or see and one of the funniest.
Older Than Steam: The gravedigger scene in Hamlet. Also his roundabout explanation as to where Polonius' body is could be seen to fall under this trope:
King Claudius: Now Hamlet, where's Polonius? Hamlet: At supper. King Claudius: At supper! Where? Hamlet: Not where he eats, but where he is eaten: a certain convocation of politic worms are e'en at him.
The "music with her silver sound" scene in Romeo and Juliet. Juliet's entire household, including her parents, her nurse, and her would-be husband, is in mourning because they think she's dead (and, watching the show, you know that she's going to end up dead soon, anyway.) The musicians who have come to play for her wedding realize that this is kind of an awkward time to be hanging around, and they try to exit discreetly...only to run into Peter, the nurse's comic manservant, who wants them to play a song to cheer him up. They try to explain to him that it's really no time to be playing music...only to have him threaten them at knifepoint.
First Musician: What a pestilent knave is this same! Second Musician: Hang him, Jack!
Little Shop of Horrors has plenty of this, since Audrey II is both horrifying and hysterical.
Avenue Q. About three across-the-line jokes per song. Assuming the line is pretty far away from "tasteful". "The Internet Is for Porn ..."
Sarah Kane's play Blasted takes this trope one step further: Ian, one of the main characters, eats a dead baby. He is also a racist, alcoholic rapist who has had his eyes eaten by a soldier who raped him with a gun.
Despite (or maybe because of) the fact that 13's target audience is teenagers, it uses a lot of this. Archie, who has muscular dystrophy, is the subject of many terminal illness jokes.
The classic Punch and Judy puppet show, especially in its harsher incarnations.
Every single production by Pittsburgh-based theatre company Rage of the Stage falls into this category, including a The Wonderful Wizard of Oz adaptation featuring an insane and heavily medicated Dorothy, a heroin-addicted Scarecrow, and a sex-obsessed Lion.
Shakespeare's Titus Andronicus is frequently interpreted as this — which may actually have been the point to begin with.
Anton Chekhov wrote comedies so dark a lot of people assume them to be straight tragedies. One of his plays famously opened in two Russian theaters at roughly the same time, one billed it as a satirical comedy, the other as a tragedy.
Pokémon: The Mew-sical has a lot of this; edgy humor is played for laughs to the point that there's a running gag of Giovanni shooting people.
Much of A Gentleman's Guide to Love and Murder is spent watching the protagonist come up with increasingly wild ways to murder his relatives, all while singing about it through operetta-style songs with delicious wordplay.
Charles Wood's H, Being Monologues at Front of Burning Cities depicts the Indian Mutiny of 1857 as a boisterous comedy, playing battle scenes, grisly executions, casual racism and even rape for laughs.
Firebugs is a satirical comedy about a town terrorized by arsonists who trick people into letting them into their homes. Two of these arsonists dupe the main character into helping them, even giving them the matches to set his own house on fire.
Anne of Green Gables: The Musical has a few examples, most notably Anne's shockingly gruesome explanation for why Avonlea's roads are red, and schoolteacher Mr. Phillips' creepy solo which is essentially one big double entendre for taking sexual advantage of (and impregnating) his female students.
Rumors by Neil Simon opens with an apparent attempted suicide, several characters are injuried, and divorce, adultery and criminal activities are crucial to the plot.
Paul Zindel's play The Effect of Gamma Rays on Man-in-the-Moon Marigolds, which ends with the protagonist's mother having a psychotic breakdown and killing her daughters' pet rabbit. The natural reaction to such a description is "Dude, Not Funny!" — but the way it's written, it is one of the funniest scenes in the entire play.
