Love Never Dies: It turns out that Christine loved the Phantom more all along — her son Gustave is the product of a one night stand they had before she married Raoul — and Raoul accepts this. Unfortunately, Meg Giry, driven mad by the Phantom never paying attention to her and being unaware of how much she did to build his fortune in America (she was forced into prostitution by her mother) tries to murder Gustave. When the others confront her, Christine is accidentally killed...but she dies in her true love's arms and Gustave is ready and willing to accept the Phantom as his father, so he is no longer alone in the world.
A Chorus Line: Paul is injured with a 50/50 chance of ever being able to dance again. And only some of the dancers will be accepted into the chorus line; the others have to go. In both cases, such is the reality of show business.
Wicked. Even though the protagonist survives, and Glinda banishes both Morrible and the Wizard, Glinda is still left alone believing every close friend she's ever had to be dead. Fiyero, now cursed as a scarecrow, and Elphaba might have survived but can never return to Oz. Glinda and Elphaba come to terms with the irreversibility of fate and circumstances and ultimately realize that the fact that they met and changed each other was what really mattered.
At least it's not so much of a downer ending for Elphaba and Fiyero compared to the book(s).
In the musical Spring Awakening, Melchoir finds that his pregnant friend, Wendla, is dead. Racked with guilt, he attempts to commit suicide only for her ghost, and his friend's ghost, to appear and convince him to live.
Even more creepy, in the original play. Mortz's ghost finds that he rest with a pride and a smile on his face, while Melchior, remorseful of his part in Wendla's death, lives on under the works of the Masked Stranger.
1776 ends with the Continental Congress signing the Declaration of Independence-with the knowledge that they have a long, arduous war that they have little chance of winning to look forward to, considering their army is a shambles and the British Navy was the greatest in the world. Hindsight tells us they'll win eventually (since it is based of history), but they don't know that.
The climax of Shakespeare's Henry V sees the vastly outnumbered English defeat the French at the (in the play) decisive Battle of Agincourt. The victorious Henry charmingly woos the French Princess Katherine to seal the deal. But the Chorus reminds the audience that "this star of England" would die just a few years later, and his son, unfit to rule, would lose France and "make his England bleed."
Although it's usually considered tragic, Romeo and Juliet finishes with the ending of a terrible feud which was probably going to have killed a lot more than five people over the years had it been allowed to continue.
Nathan of Thrill Me does get his parole, and so gets to go out and live his life. However, there are only two characters in the show, and this makes it look like the only meaningful relationship of Nathan's life was with Richard, who is now dead. The ending can be played better or worse—anywhere from Nathan still being fully obsessed with Richard to Nathan finally beginning to get over him—but regardless, Nathan gave up most of his young life to a bad relationship, and only now gets any chance at starting over. Not to mention the fact that a convicted murderer being set free isn't necessarily the best ending.
Fuddy Meers: Claire remembers all that happened before her condition until she falls asleep in the car at the very end.
The Pillowman: Katurian is executed despite his innocence, Tupolski gets away with it completely, and Ariel is ordered to burn Katurian's stories. He keeps them instead.
Ramona: In the current version of the Ramona Pageant, Felipe manages to save the rancho from foreclosure and convince Ramona to marry him, but Alessandro is dead, as is Eyes of the Sky.
"Theatre/Hamilton": Aaron Burr shoots Hamilton and gets his revenge, only to immediately realize that the world was "wide enough for the both of us" and that he is permanently doomed to be history's villain forever. Unlike most of the other Founding Fathers, Hamilton runs out of time and dies too young to ensure his legacy. However, he gains the post-mortem respect of arch-nemeses Thomas Jefferson and James Madison, and Eliza works tirelessly to cement his earned place in American history and to accomplish all the things her husband never got the chance to. She succeeds.
Les Misérables: Jean Valjean dies, as did a good number of the main cast. However, unlike most of the other casualties, who were either murdered, committed suicide, or died in misery and squalor, he dies with Cosette and Marius by his side, and is welcomed into Heaven with open arms by Fantine to a triumphant reprise of "Do You Hear The People Sing?". The song even highlights his contentedness at being able to learn how to love after spending a significant portion of his life mired in fear and/or hatred.
Fun Home's musical adaption: Bruce kills himself, but Allison is able to come to terms with his death and moves on, completing her autobiography with a picture of him catching her in a pool.