Put simply, in the ancient Greek play Antigone, the title character wants to bury her brother, against the wishes of her uncle the king. In ancient Greece, they would see Antigone as caught between two horrible options; not honoring the dead, or defying her rightful ruler. Thanks to liberalism, individualism, feminism and the separation of church and state, a modern reader would see Antigone as rebelling against a corrupt and authoritarian state, with the only problem being the possibility of getting caught doing it.
However, because Antigone seems to be unquestionably doing the right thing to modern eyes, modern performances of the play usually shift the focus to her uncle, and instead emphasize his two horrible options; condemning his niece and nephew to dishonorable deaths as is required by the laws of the city, or placing his family above those laws by burying his nephew and sparing his niece.
Most scholars will tell you that this is how the play is meant to be interpreted. Remember, it is the Chorus that is supposed to embody the focus of the piece, and both the Chorus and the play itself spend a great deal of time explaining Creon's predicament and the possibility of an impending invasion, the implication being that if he appears weak and emotional, he believes the state will be weakened and fall.
The Nazis sympathized with the plight of the uncle more than they did with Antigone, in fact, renaming the play after said uncle, Creon. Presumably it was for opposite reasons.
The lighthearted treatment of rape in Ancient Greek and Roman comedies can make it impossible to enjoy them. Terence is probably the most jarring, in that he makes it clear that his heroes are violently and traumatically assaulting his heroines, yet they still end up together at the end, with the man excusing himself by revealing that he was drunk, or that he thought she was a slave.
Also the casual way slavery is dealt with, with a (serious) threat of killing or whipping a slave treated as comic.
Most modern productions of Shakespeare's Much Ado About Nothing leave out one of Claudio's lines near the end of the play. To make amends for his part in Hero's supposed death, he agrees to marry her cousin, but is told that he can't see her face until he swears it before the friar. He replies that he would take her in marriage even if she were "an Ethiope" (that is, a black African).
Joss Whedon's modern day adaptation not only keeps the line in, but plays it for laughs with a black woman giving a Death Glare while Benedick sighs at what an idiot his friend is.
"Ethiop" seems to have been a favourite insult of Elizabethans — Lysander calls the dark skinned Hermia this in A Midsummer Night's Dream. This could likely be a Values Dissonance in regard to food culture as well, as Europeans almost never ate raw meat when they could avoid it, save for a few backwoods hermit-y types. Therefore being compared with someone who ate raw meat (look up Ethiopian dishes sometime), which is something not even the lowest commoner would sink to, would be seen as equally vulgar an insult.
Many modern readers/watchers are much less willing to forgive Claudio, who was ready to have Hero put to death. While today cheating the night before your wedding is considered a pretty terrible thing to do, being executed for it sounds way too extreme, and the fact that Claudio doesn't even investigate to discover the situation makes him seem shallow and fickle, by no means deserving of Hero's hand in marriage. His response would be been considered perfectly reasonable to Shakespeare's audience, however.
Everyone remembers Othello for the (then) controversial interracial marriage of its hero and Desdemona. Nowadays people are far more likely to take issue at the fact Desdemona can only be sixteen at the very most... and Othello's roughly the same age as her father (who was once his friend). It's disturbing that so many characters speak so lustfully about her, considering how young she is. At the time, a 16 year old would have been considered fully adult and capable of raising a family. Indeed, in much of the world, that's still the case today. Furthermore, Desdemona is described as a competent and mature young woman who has been managing her father's household for years — no sheltered innocent.
Juliet is thirteen. Even in the sixteenth century, that was so young that a lampshade is hung on it in the play: her father originally thinks she's too young to marry and her suitor Paris will just have to wait... for three years. In some adaptations (most notably the 1968 movie), the reason her father was hesitant was because it was implied that he married Juliet's mother too young and had grown to regret it. Juliet's mother comments at one point that she was about Juliet's age when she was married and started having children.
The play has a happy ending, as the villain has been forgiven for his attempted judicial murder and has even become a Christian, thus giving him the chance to go to Heaven. At least, that is what the original audiences would have thought. Modern productions are more likely to sympathize with Shylock: the Royal Shakespeare Company once put on a production where most of the cast were dressed as Nazi stormtroopers. (One critical essay pointed out that for a woman who speaks so movingly about "mercy," Portia is a vindictive bitch: she forces Shylock to renounce his religion and give his property away to the daughter who betrayed and stole from him.)
It's actually Antonio who makes those demands of Shylock and the Duke who forces him to accept them. Portia is certainly vindictive in the sheer amount of rubbing salt into the wound (pardon the expression) after she's found the loophole that gets Antonio off the hook, but after that she takes a back seat to pretty much everyone else who instantly start ganging up on Shylock.
Portia tricks her boyfriend into making it look like he had betrayed her (by making him give her a ring she made him promise to never give to anyone else while in disguise), then pretends to act like he had cheated on her. This is played for comedy, rather than as an indication that she is psychopathic. She's also quite racist to her two suitors that come before Bassanio and nothing is made of it. She dislikes the Prince of Morocco simply for his dark skin and when he chooses the wrong casket she says "may all of his complexion choose me so". With the Prince of Aragon her dislike is a little more justified since he is indeed vain and arrogant and England was in the midst of the Armada so of course a Spanish character wasn't going to be likeable.
One ingredient in witches Brew in Macbeth is "liver of blaspheming Jew." Eek! The line is usually left out of modern productions.
Shakespeare's The Taming of the Shrew is divisive on whether it is dissonance—some people perceive the ending monologue and summation of the play, in which the female character declares all women must be subservient to men in a literal sense, while others assume it is meant to be satire or sarcasm. This debate comes up almost as often as whether or not Romeo and Juliet are meant to be romantic and tragic, or just tragic.
The Duchess of Malfi revolves around a forbidden marriage and what we would nowadays consider to be an honour killing. While her behavior in disobeying her family, marrying her steward, and actually proposing to him rather than vice versa, would have met with strong disapproval from most audiences, Webster is clearly depicting her as the most noble character in the play, the only one who didn't do anything seriously wrong; the rest of the court is populated by scheming tyrants, incestuous brothers, hypocrites, and murderers — the anti-hero protagonist is a killer-for-hire. This was a very radical play when it premiered. Nowadays sympathies are entirely with the lovers.
The Magic Flute: To the extent that the opera has An Aesop, it's about how you shouldn't trust or even listen to women, and how women need a man to guide them lest they become too uppity. (On the other hand, it doesn't hurt that all the parts about the brave and noble men overcoming every challenge are complete snooze fests, while the villainous Queen of the Night gets the twobest arias in the whole opera, including one of the most famous in the entire genre.) With a side of "black men are too ugly to get any, so they'll resort to raping white women, to whom they are irresistibly attracted."
On the other hand, Pamina is initiated with Tamino. Considering that both Mozart and Emanuel Schikaneder (who wrote the libretto) were both Freemasons and that the opera is full of Masonic themes, and that to this day most Masonic lodges do not initiate women...
My Fair Lady was based off George Bernard Shaw's Pygmalion, and the original ended with Eliza going off to marry Freddie, not returning to Higgins. Subsequent versions changed his play's ending to one similar to My Fair Lady. Shaw was so upset with the people who changed the ending that he wrote an essay explaining why Eliza and Higgins would never end up together, and why Eliza would be happy with Freddie (though they would experience a financially difficult marriage).
When it was first made, Eliza came across as much more unacceptably uncouth to theatre-goers, and therefore just as bad as Henry, whereas it's getting more and more common to see Henry as a misogynistic villain putting Eliza unfairly down. They're each supposed to be a Jerk with a Heart of Gold, but current values don't look favorably on characters like Higgins.
A Doll's House: In contrast to many of Ibsen's other plays, this one has a straighter example of this trope — when Nora leaves her husband, she leaves her children behind as well. At the time, the concept that men had automatic custody rights to any children from a marriage was completely natural and that particular decision wouldn't raise an eyebrow. To modern audiences, this is much less natural and has levelled charges of irresponsibility on the guilty party. To be fair, Nora probably wasn't necessarily leaving the children to their father, but rather the governess. An earlier scene indicates that Nora deeply trusts her children's governess (who was also her governess too) to become the maternal figure should anything happen to Nora.
In a couple of his plays, French playwright Georges Feydeau has English-speaking characters in Funny Foreigner roles often speaking a not very accurate gibberish which while hilarious to the contemporary audience doesn't hold up well in translation. Translation Convention is to either to have those characters speak the same English as the French characters but to be not understood by them, or else to adapt them into Funny Foreigners from other countries.
Gilbert and Sullivan's The Mikado has some of this: a few songs use the word "nigger", which is changed for modern productions (there's a long-standing tradition of changing the lyrics to G&S songs anyway). Many have criticized the operetta for making fun of the Japanese, but it is almost certainly meant to be a satire of British society.
In the 1920s musical "No, No, Nanette", Nanette is a young woman old enough for her sweetheart, Tom, to be begging her for marriage. When she wants to take a trip to Atlantic City (to a cottage that her own family owned, no less), her adopted mother refuses to let her go, on the grounds that it's inappropriate, saying that keeping her in the house and training her to be a proper lady until she gets married is for her own good. Tom also finds the idea very distasteful (though given that his objections center around Nanette hanging around with strange boys, jealousy probably was a factor). Even when Nanette's adopted father thinks the two of them taking a vacation to the cottage sounds fine, he still insists that she needs a chaperone.
The ending of Annie Get Your Gun, in which the female main character throws a target shooting competition and gives up a successful show-business career in order to win the heart of the man who was jealous of her success, is a classic example of something that seems outrageous today that would have seemed completely reasonable when it was written. In Real Life, the opposite happened: Annie Oakley's husband gave up his sharpshooting career for hers.
On the other hand, however, Annie wasn't always the loud and brash sharpshooter as she was portrayed in the musical. If anything, she was actually a very quiet girl who frequently did needlepoint in her spare time.
Annie Get Your Gun was written deliberately to be post-war propaganda, to lure women out of the factories and back into the kitchen.
Revivals have Annie throw the contest, but Frank finds out. He's touched that she would give up her career for him, apologizes for the way he was treating her, and they live "scappily ever after."
The musical Carousel features a defense of domestic violence. Julie, thinking longingly of her abusive dead husband, remarks wistfully that "it's possible for someone to hit you ... hit you very hard ... and not hurt at all." The audience isn't supposed to cringe at how cowed she is, but to sigh over this romantic moment.
Similar to the above, Nancy's staying by Bill Sikes even in the face of his abuse in Oliver! comes across as overly submissive and lacking regard for her own well-being to modern audiences, but there were no abuse hotlines in Dickensian London.
In The Laramie Project, one of the interviewees is a straight stage actor discussing how he once played the lead role in Angels in America, but his parents refused to attend the play because they didn't want to see him play a gay man. However, he also played the title character in Macbeth back in high school, and they were right there in the front row as he portrayed a mass murderer.
The Asian Speekee Engrish stereotypes used at the climax of (some versions of) Anything Goes to stop Hope and Evelyn's wedding and to hook up Hope and Billy and Evelyn and Reno have REALLY not aged well—to the point where they're often cut out altogether.
In her article "Shakespeare in the Bush", Laura Bohannan describes recounting the story of Hamlet to a pre-literate tribe of Bushmen in Africa. Their society had no concept of ghosts and it was proper custom to marry your dead brother's wife. The story is still a tragedy, but has become a tale of powerful witches who send false visions and murder their own sisters.
The Children's Hour is about two female teachers accused of being together by a bratty Enfant Terrible student, and how it ruins their lives. It comes off as very much a Period Piece nowadays as few would bat an eye to them being a couple. Karen cheating on her fiancee still wouldn't be socially acceptable but it wouldn't create the huge drama it did in the play, especially to the degree of getting into a court case about it. It's even worse in the 30s film adaptation - a scuffle between two women dating the same man wouldn't cause such reactions at all anymore
Edouard Bordet's 1920s play "La Prisionerre" (The Captive) tells the story of Irene, a wealthy twentysomething in love with a somewhat older woman. Pressured by her father (who seems to be onto her) and frightened of her feelings, she marries a male childhood friend who's long been infatuated with her. Act Two depicts their loveless marriage; Irene has reconnected with her old lover and her husband is also cheating with an old girlfriend of his. The play ends with Irene and her paramour going off together and her probably-soon-to-be-ex-husband wanting to marry his mistress. Sounds like a perfectly happy ending - but at the time, it was considered a tragedy because Irene was "lost" to homosexuality.