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.
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.
Goethe and Schiller, thanks to being the quintessential German writers and often mentioned in one breath (they were Heterosexual Life-Partners for most of their careers) can come of as extremely stuffy with their plays a bit formulaic and in the case of Goethe's Faust the same problem as with Hamlet above: Just a bunch of quotes strung together.note If you hear a famous turn of phrase in German and have to guess where it's from you have a better than even chance it ultimately comes from Luther's translation of the bible or Goethe's Faust However, back in the day when plays like Die Räuber (Schiller) or Götz von Berlichingen (Goethe) were first performed, they were downright revolutionary. Some of this revolutionary zeal could be seen centuries later, when the line "Geben Sie Gedankenfreiheit" (give freedom of thought) from Schiller's Don Carlos was met with roaring applause during a performance in the GDR because most of the audience could not help but notice its appropriateness for their oppressive regime.
The "first" plays to break the Aristotelian unity of place, action and time described in Poetics. note Even some classical Greek drama, mostly be Euripides, actually broke this convention, Aristotle was wrong on this if it is interpreted as a descriptive text, whether he was right on it as a prescriptive text is for the audience to judge. Aristotle was regarded as pretty much right about everything for most of the medieval period and the Renaissance had a fondness for everything Greco-Roman. Going against that took guts. Today plays set in different places over several days containing numerous plots and subplots are par for the course on the stages of the world.
This was a far bigger deal in Continental Europe, especially in France. In England, during the Elizabethan Age, Aristotle and classical Greek texts were not as dominating an influence as Roman drama by Seneca and Latin texts. So William Shakespeare happily violated the unities, likely because he didn't even know about it to start with. The most learned and informed dramatist, Ben Jonson, gently ribbed his friend in the First Folio for his "little Latin and less Greek" and his comedies and dramas were the most formal and classically structured.
Shakespeare was unpopular in France until the Romantic era for his violation of classical unities, with Voltaire dismissing him on these grounds. It took the critic Samuel Johnson to first defend Shakespeare's approach as valid and argue that the unities are more guidelines than actual rules. In Germany, the Sturm-und-Drang avant-garde saw Shakespeare as a modern writer on these grounds and admired his bold original spirit. In France, Victor Hugo a huge Shakespeare-buff wrote a play called Hernani that was a scandal in its day because it violated the "classical unities", which had underpinned France's Golden Age of Cornielle, Moliere and Racine.
Bertolt Brecht defined his conception of drama as "Non-Aristotleian" at the start of the 20th Century. By the time he arrived, drama had already greatly advanced away from the original Aristotleian schema (Anton Chekhov and Henrik Ibsen were the innovators on this front) but Brecht noted that most people still agreed with ideas of three-act structure and the division of epic form and dramatic form, a large cast and the small cast, and the idea that a tragedy can begin end and achieve catharsis in a small space and time.