West Side Story. Today this musical seems like a terrible conglomeration of clichés on top of the material it takes from Romeo and Juliet (which itself was a fresh take on a clichéd story in its day). But West Side Story started a lot of musical conventions which became clichés, including its (for the time) grittiness, its use of street slang and cursing, its (relatively) sympathetic portrayal of minority characters, and its use of ethnic musical conventions.
West Side Story has arguably aged very well, particularly since its characters look a lot less stylized and stereotyped than most 1950s "delinquent" characters. In fact, it probably influenced street-gang fashions (particularly the wearing of the bandanna) for decades afterward!
William Shakespeare. From the introductory paragraph to chapter 6 of Lawrence Lessig's Free Culture:
Henry V is particularly susceptible to this, as it's been mined, deconstructed, or outright stolen from for basically every war movie ever made.
In fact, it's a common joke amongst theater folks: a woman (for some reason, it's always a woman) sees Hamlet for the first time and complains, "I don't know why people make such a big deal about it. It's just a bunch of quotes strung together."
Hamlet has been praised as "the crowning achievement of Elizabethan drama" so many times that it's now easy to forget what a unique play it was in its day, and how revolutionary its approach to drama was compared to other plays of the Elizabethan era. At the time, it was a pretty damn big deal for a play to consciously fall so far on the "Character" end of the Sliding Scale of Plot Versus Characters, spending just as much time examining its title character—his obsession with death, his crushing emotional uncertainty, his relationships with his family, etc.—as it spent on the revenge story at the heart of the plot. Hell, the fact that we even have a Sliding Scale of Plot Versus Characters is arguably thanks to Hamlet's influence.
Oklahoma!: Broadway musicals like this one may seem quaint, dated, and silly now, but compared to the typical showgirl fare of the time, their integration of music, dance, and plot, as well as their darker themes, were ground-breaking. Both Show Boat and Oklahoma! were written by the same librettist, Oscar Hammerstein II. Whichever show one chooses to credit, Hammerstein was instrumental in this development of a kind of musical based more on narrative and character than entertaining numbers. And without Hammerstein there would certainly have been no Stephen Sondheim, who took that development even further. Sondheim has pointed this trope out as well (Allegro is another, less well known, Rodgers and Hammerstein show): "People don't understand how experimental Show Boat and Oklahoma! felt at the time they were done. Oscar is not about the 'lark that is learning to pray' — that's easy to make fun of. He's about Allegro."
Revivals of the play these days take this into account by trying to make you forget it's a play at all, performing it in the open instead of on a stage, making it more like a "happening" and thus preserving the original spirit.
Bürgerliches Trauerspiel (Bourgeois Tragedy). During the Age of Enlightenment this sub-genre of drama arose, in which virtuous commoners were shown as victims of the machinations and depravities of aristocratic villains, which at the time was considered daring and subversive, sometimes even seditious and revolutionary. Some of them are still performed today, most notably Gotthold Ephraim Lessing's Emilia Galotti (1772) and Friedrich Schiller's Kabale und Liebe (1784), but are often now seen as dated and quaint. This is not an entirely new trend, as the bourgeois values propounded in "bürgerliche Trauerspiele" became subject to criticism themselves, which in the 19th century led to the writing of Realist dramas with bourgeois villains.
Could be called "The Problem With Chekhov." In Chekhov's day, naturalistic theater about people's real emotional lives was a strange and radical notion. Now it's what almost every play is about, and it's hard to understand why Chekhov's work was so powerful at the time. In fact, Chekhov plays themselves can sometimes seem as stolid and old-fasioned as the works he was rebelling against at the time.