Is it permissible to mention a funny situation or a joke from a movie or play that includes illicit content, or is that considered an invitation to illicit acts?

Mentioning a humorous anecdote or situation that occurred in a play, while also naming the play, could lead the listener to seek out and watch the play, which is impermissible. It might also foster affection for the actors or trivialize what is objectionable. However, if the speaker is certain that these dangers are absent, then there is no harm. As for mentioning a humorous anecdote in isolation, without stating that it is from a play, and provided it contains nothing objectionable, there is no prohibition.