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Stella in A Streetcar Named Desire. She's basically put between a rock and a hard place, forced to side with either her psycho, gold-digging, mooching sister Blanche or her boorish, abusive husband Stanley. Even after Stanley hits her while pregnant and later rapes Blanche, she ends up having to side with Stanley.
Arguably the original Butt Monkey can be found in the character of Pedrolino from the Commedia dell'Arte. Pedrolino is eternally in unrequited love, and is also the butt of every available joke. In family Commedia troupes, this role would be given to the youngest member of the family.
Kilroy from Don't Drink The Water is such an annoying kiss-ass that there's a Running Gag throughout the second act of him getting horribly injured: he slips and breaks his arm, gets hit in the head with a brick, causing him to go crazy, and to top it off Walter accidentally shoots him!
The Duchess of Malfi. Poor girl. All she wanted was to get married... and look at the horrors that unleashes! Imprisonment, mental torture, her eventual murder... Her hapless husband Antonio also applies. Malfi probably has the earliest instance of the hitman being something of a Butt Monkey too.
Papageno from The Magic Flute. Lips sealed with a magic lock; coerced into the initiation rites for an organization in which he has no interest; and taunted with the contrived inaccessibility of his love interest until he literally attempts suicide. To be fair, Papageno brings the magical lock thing on himself as punishment for lying, but throughout the opera the punishments for his actions tend to be disproportionate.
Miss Saigon: Kim. Orphaned when her parents and other family members are killed in an attack, forced to work as a prostitute to support herself—this is how she loses her virginity—has the love of her life all but literally ripped away from her during the Fall of Saigon, has to endure pregnancy and childbirth all on her own, lives in poverty, has to kill her cousin when he tries to kill her mixed-race son, flees to Bangkok, finally thinks she's going to be reunited with her lover after three long years (and this belief is basically the only thing that has sustained her all this time), only to find that he's married and his wife doesn't want her or the kid around, and finally kills herself both out of grief and to force them to take the child to America with them. Oh, and by the time this is all done, she's all of 20-21 years old.
In The Most Happy Fella, Herman gets pushed around a lot by Pasquale and the other workers, but he has no complaints until the last scene.
There are dozens of Shakespearean examples. The most extreme would be the entire cast of Othello, not just its criminally naive hero (one character, the perpetually stupid Roderigo, is even referred to in the cast list as "a gulled gentleman"). Iago, revolting creep that he is, grins as he puts them all through hell.
Another major example would be Shylock of The Merchant of Venice. He's forced to endure endless abuse from the Venetians, his daughter runs off with a Christian and steals his dead wife's ring, he's forced to convert and give up his profession in a trial that's completely unfair... Yes, his wish for a pound of flesh is villainous, but Values Dissonance has made this alleged "comedy" almost unwatchable.
Malvolio in Twelfth Night. He's a haughty Puritan who doesn't like roistering, it's true, and has a daft infatuation for Olivia, but being mistaken for mad and incarcerated leaves something of a bitter taste in modern viewers' mouths. Another Butt Monkey of epic proportions is Sir Andrew Aguecheek, Upper-Class Twit extraordinaire. Dense, naive and good natured, with absolutely nothing going for him except for his money, it's possibly a good thing he's so stupid as he never seems to realise the contempt everyone else holds him in. He tries to woo Olivia with no success, his 'friend' Sir Toby is forever sponging off him... He can't even get a challenge to a duel right!
In The Taming of the Shrew, everyone at least breaks even, except for Hortensio, who gets beaten up by Kate and ends up marrying a widow who hates him, but needs a husband to be reaccepted into Paduan society.
Ophelia in Hamlet (although almost everyone in that play could qualify).
In the ballet Fancy Free, one of the three sailors always gets conned by his two buddies into having to pay for everything at the bar.