In the Broadway (now off-Broadway) musical Avenue Q, Gary Coleman is depicted as a character. The misfortunes of his life are mocked extensively; he actually works as the building super in the show and, at one point, sings about the fact that his purpose in life is to bring happiness to others via schadenfreude, or making others happy that they're not him. All of this is slightly cringe-worthy given Gary Coleman's sudden death.
In the Hungarian version of the same musical, Gary Coleman was decided to be too unrecognisable to the Hungarian audience, so was replaced by Michael Jackson. As can be imagined, there was originally a joke about him moving to Avenue Q after losing his fortune to a lawsuit from a pair of 5-Year-Olds. After his death, however, the joke was changed to a joke about his spending too much, and thus faking his death.
Aristophanes' play The Clouds poked fun at Socrates. A few years later, Socrates was executed for pretty much the exact things Aristophanes made fun of, even though some of them were things which Aristophanes made up for Rule of Funny.
In Crimes of the Heart, middle sister Meg has been lying to her grandfather about how successful her singing career is (it's not). After a particularly good evening, she is so giddy that she resolves to tell him the truth - "And if he can't take it, if it sends him into a coma, that's just too damn bad." Guess what happened to ol' Granddaddy overnight. Played literally in that Lennie and Babe can barely tell Meg the news because they are laughing hysterically. Black comedy, indeed.
The Vagina Monologues features one monologue wherein a girl is raped by her father's friend, her father kills the guy, and her mother won't let her father see her anymore. Then her mother took her to an older woman who taught her to masturbate. After Abu Ghraib, Eve Ensler admitted she couldn't see women as brutalizers before that.
Billy Elliot has a good example in the song "Merry Christmas, Maggie Thatcher", which contains the lyrics "We all celebrate today,'Cause it's one day closer to your death.". Upon the day of the death of Baroness Thatcher it was put to an audience vote as to whether the song ought to be performed or not and the previously humorous song seems to many to be in bad taste.
Cirque du Soleil's KA has some black humor in its No Talking Or Phones Warning when the "audience member" who breaks the rules is pushed into a seemingly bottomless pit (that the show's moving stages emerge from) by the villains. On June 29, 2013, a performer fell 50+ feet into that space during the climactic Wire Fu "Battlefield" sequence; her resultant death is Cirque's first onstage demise. Between the investigation into the disaster and the Harsher in Hindsight concept of a show where characters are constantly tumbling into the void (sometimes to their doom), the show went on hiatus for a few weeks before reopening without "Battlefield".
French theater celebrity Molière died from tuberculosis a few hours after a performance of one of his work, in which he also had the main role. Which one? Le Malade imaginaire (The Hypochondriac).
Pretty much all of Rent after Jonathan Larson's Author Existence Failure (and he died on the night of the final dress rehearsal Off-Broadway.