I just rewatched the TNG episode "Frame of Mind" and I was wondering if the play that Crusher was directing (and Riker and Data were starring in) was a real play or maybe based on a real play?
Very likely, no.
According to Memory Alpha, Frame of Mind was a play written and directed by Dr. Beverly Crusher.
According to this, we can exclude that it is a real play, given that a fictional character is credited to be the author.
Background information for this episode reports that it was written as an answer to "What if Riker woke up in an alien insane asylum and had no idea how he got there and was told he was crazy?".
Given this information, we can also rule out it being based on a real play, because it seems that the actual play is nothing more than a plot device to put Riker in the intended situation.
Anyway, Brannon Braga, the episode writer, said that some influence can be traced to Roman Polanksi's Repulsion, one of his favorite movies.