WARNING: Nearly every example is a spoiler. Read at your own risk!
To William Shakespeare, a "tragedy" meant "play with a tragic ending for the protagonist", so most of his tragedies count. (Romeo and Juliet is the best example) but not all, seeing as plays with a Villain Protagonist counted too. (Many of them, like Macbeth were tragic because the story involved a Fallen Hero who was corrupted by evil, and while the endings were rarely all-too happy, they still ended with a better ending than most examples here.)
Cabaret. The Nazis are in power, Cliff flees Berlin and what looks to be his only chance at a family, Sally has a mental breakdown, and the Beta Couple falls apart for the very practical reason that one of them is Jewish. Oh, and in some productions, it's implied that the Emcee ends up as a victim of the Holocaust (Usually with the blatant implication that it was because he's gay). Musicals are such light-hearted fun, ja?
The movie version is slightly less of a downer, with Maximilian admitting to be a Jew, so the Beta Couple can be married; but being Jews in Nazi Germany...
Chess (at least, the two most well-known versions of the plot) ends with Anatoly returning to the Soviet Union with the assumption that it will get Florence's father released from prison, when in reality, we learn, Anatoly's mostly being exchanged for some American agents and Florence's father may not even be alive.
Wicked. Yeah Elphaba and Fiyero are alive, but they have to leave their homes, he's stuck as a scarecrow, Glinda is now all alone with every single person she's ever been close to dead or in jail, Nessa is dead, Boq is a tin man and the Wizard had his world shattered by Glinda's revelation. Irreversible.
Even worse, she has to act happy most of the time, in both the book and musical.
Ha, you think thats bad, in the novel version, Fiyero is killed off half way, Elphaba goes partly mad with grief and depression, calls Glinda a traitor and abandons her, all her animal companions are killed off, she fails to rescue her half-niece from the Wizard, kills her Nanny, and then accidentally get killed by Dorothy and the wizard gets to fly away happy. The worst part is Elpheba is forever remembered afterwards as the "wicked witch", eternally being remembered as the personification of evil.
The Wizard flies away, but not happily. He gives up the magic book he'd been seeking his whole life when he learns that Elphaba was his daughter, so in the end, Elphaba did accomplish her life's mission of keeping the Wizard from the book and, indeed, bringing the Wizard down. She just wasn't alive to see it.
Bousille et les justes. You know what happens? The main character, the only good character in the story, is forced to testify for a guy, knowing that it's wrong. AND, he's incredibly morally opposed to lying. In the end, he feels so guilty about his actions that he hangs himself in his garage. Ouch.
Though, one may argue that this is worse, seeing as Juliet ddn't have to live with the crippling grief of losing her love for long...
Spring Awakening: Wendla dies from an abortion her mother forces her to have after indirectly getting her knocked up by sheltering her from sexual education at the beginning of the play. Moritz kills himself rather than face the shame of confronting his parents after flunking out of high school. Authority is deadly and free will has its price and needs control or else.
Let's not forget the fact that Melchior found out that Wendla was dead when he went to meet her in the cemetery and saw her name on a tombstone.
Oliver!, considering the only one who has a happy ending is Oliver himself. And that's after watching Nancy killed by Bill Sykes, who is then shot (or in the film version, as in the original book, accidentally hangs himself) trying to escape with Oliver. Must've been pretty traumatic to be a part of...
Hair. "Let The Sunshine In." We could use it now that Claude has died in Vietnam.
Miss Saigon. Unable to convince the father of her child and his new wife to take her son back to America with them, the titular character shoots herself in order to force their hand. (As a half-Vietnamese child, he would have been an outcast in Asia.) The father is left with the dead woman in his arms, lets out a gut-wrenching cry of anguish... Curtain.
Fiddler on the Roof: Perchik and Hodel are in a prison encampment in Siberia. Chava has been disowned by Tevye after she ran off with Fyedka, a russian soldier. Those don't even hold up to the true Downer Ending, when the entire population of Anatevka is forced out of their town by Mother Russia. Cue a sad ending where we see the entire town packing their bags and leaving home depressed. The last thing we see is Tevye cuing the fiddler to stop playing.
Although Tevye & Co. are planning to come to America, which will get them out of Europe just in time to avoid WWI, WWII, and the Holocaust.
But not Tzeitel and Motel, who are going to Poland.
And then there's Chava and Fyedka, who are on their way to Krakow.
A Streetcar Named Desire. Sure, everyone knows Blanche's famous last line "I have always relied on the kindness of strangers." In the play, these are her final words as she is led to an insane asylum. After being raped by her sister's husband. And her sister blames her. And the sister and husband stay together. The kindness of strangers indeed...
Not even Blanche can be called a sympathetic character in this one. She drove her family's estate into foreclosure, made off with several expensive clothes like a bankrupt company's CEO and his golden parachute, stayed in a bordello after that, then comes to New Orleans to mooch off her sister, immediately accusing Stella of abandoning the family and calling Stanley racial epithets before she even meets him. While Stanley's ultimate actions can't be justified, they do come from the very real belief that Blanche will drive his family and his best friend into the same pit of ruin. Tennessee Williams seemed to have a thing for plays with Downer Endings and few sympathetic characters. See below...
The film version ended on a slightly more hopeful note, at least regarding Stella's relationship with Stanley. While in the play, Stella is broken over the whole affair and is sobbing at the end while Stanley unbuttons her blouse and prepares to have his way with her, the movie ends with Stella finally telling Stanley off, taking her baby and leaving the apartment, saying that she'll never go back to him again. Even though the audience knows she probably will go back, it still makes for a nice "You go, girl!" moment.
The Glass Menagerie. Tom is essentially forced to make a very selfish decision and abandon his mother Amanda and sister Laura rather than spend his entire life working a job he hates and taking care of a crippled and emotionally disturbed sister and an overbearing mother whose ways are too stuck in the past to help anyone.
Come on. The part where Jim the gentleman caller leaves Laura after saying that he's engaged (after that KISS with Laura) is itself a downer in some ways.
One of the four movie versions deviates this to a Bittersweet Ending, as while Tom still leaves, Laura is shown to have overcome her reclusiveness to begin a new life.
Sweeney Todd: The Demon Barber of Fleet Street, though not exactly a happy plot to begin with, has its funny and entertaining moments in its morbidity. However, the ending pretty much takes the cake in what's also a bit of a Kill 'em All as Sweeney Todd, after murdering the Beadle and a crazy old woman before finally taking vengeance upon Judge Turpin, threatens and nearly kills his disguised daughter Johanna, then tries to find Toby (to eliminate him as well), realizes that the crazy woman he just killed was actually his wife, then kills Mrs. Lovett (by tossing her into her oven and burning her alive, no less) for not telling him that his wife was alive. The show ends with Sweeney, blindly grieving over his wife, getting his throat slit by Toby, who had just undergone a seriously horrific Break the Cutie ordeal which consisted of him learning exactly what Sweeney and Mrs. Lovett had been up to down in that bake room and then being stalked by the two in the sewers in the attempt to kill him. And then it takes the whole thing one step further by having Toby (who usually had genuinely white hair at this point) then immediately going back to working the giant meat grinder humming an Ironic Nursery Tune, showing that he has been driven completely insane by said ordeal.
Stephen Sondheim has gone on record confirming that Anthony and Johanna are the only two to receive a "happy" ending. As happy as it could be, anyway.
The musical Assassins ends with the various Presidential assassins and would-be assassins winning the battle for Lee Harvey Oswald's soul, resulting (in the strange universe of the play) in the death of John F. Kennedy. Guess who wrote the music and lyrics.
The Rise and Fall of the City Mahagonny ends with the protagonist executed and Mahagonny in the midst of collapse. The protagonist's body is carried across the stage alongside various political banners calling for such futile things as the restoration of the golden age, while the ensemble sings, "Can't do anything to help a dead man."
More to the point: he kills himself thinking the best thing he can do for his family is get them his life insurance. His suicide invalidates the policy.
Bat Boy: The Musical: Bat Boy/Edgar has a psychological breakdown and flies into a murderous/suicidal rage after it is revealed to him that Meredith is his mother and Shelley is his twin sister, Mrs. Taylor burns her last surviving child alive because she mistook him for the Bat Boy, and Dr. Parker kills Edgar, Meredith, and himself, leaving Shelley and the few surviving townspeople to tell the story to the man who was coming to take Edgar away to a mental institute.
The original ending of The Bad Seed sadly ended with the the mother committing suicide and the evil daughter surviving.
The original Vanities ended at the third act with the three characters' friendship apparently being irreparably broken in a drunken catfight between Joanne and Mary after an ill-fated reunion. The musical fixed this with a Distant Finale where the characters reunite and make up.
Les MisÚrables. Sure, Marius and Cosette end up marrying, but after 90% of the characters die. Every single student at the barricade except Marius and Valjean are killed in the shooting, and Valjean dies at the very end after the wedding (but not before Javert realizes that he's been pursuing a pure and innocent man for all this time who just spared his life, and kills himself in despair). To top it all off, out of all the characters that are alive, two of them are the Thenardiers, the truly despicable, thieving characters who get away with all the looting they did.
The Last 5 Years, an off-Broadway musical by Jason Robert Brown, has a severe downer ending. Jamie has cheated on his wife, Cathy and left her. What makes this even worse is that we knew this was going to happen from the beginning. Cathy's story starts at the end of their relationship, while Jamie's starts at the beginning. The last song contrasts Jamie's remorse over their failed marriage with Cathy's excitement at having gone on their first date. Ouch...
The Yeomen of the Guard: Yes, a jolly Gilbert and Sullivan opera. In which the jester Jack Point, the main character, a classic tenor G and S hero collapses (it is left to the audience to decide if he is dead, the libretto says 'falls insensible', but Gilbert originally planned to kill him) with a broken heart when he finally realises the Ingenue is lost to him forever. All the harsher for the audience as throughout the last act, EVERYTHING works like any other operetta, and you expect all of the romantic loose ends to be tied up cleanly as per usual. In this case they are, with the hero the sole exception...
Mother Courage and Her Children In a way, it's kind of the point of the play, to show what happens when people learn nothing. Trying to profit from war to support her family, the traveling merchant herself instead winds up with 3 dead children and a cart full of crap.
One playwright who seemed to specialize in this Trope was Eugene O'Neil; angst and evil deeds, leading to Heel Realizations and self-inflicted penance for the guilty part seemed to be the theme of most of his plays. Well known works on that subject included Mourning Becomes Elektra, Desire Under The Elms, and Strange Interlude. The only play he wrote that had a somewhat pleasant ending was Ah Wildreness, a wistful re-imagining of his youth as he wished it had been.