Any play written by Creator/Samuel eckett, with Waiting for Godot probably being the most well-known example.
For that matter, anything written by other Absurdist playwrights - Eugene Ionesco, Arthur Adamov, Harold Pinter, Jean Genet.
Waiting for Godot is, rather ironically, Beckett's most sensible piece of work. One of his other plays consists of half a minute of breathing noises that start and end with birth cries coming from a pile of garbage.
Most of Beckett's plays are downright sane compared to his play "Endgame", involving 4 characters living in an underground basement after a possible nuclear holocaust. One character is confined to a wheelchair, and his parents live in dustbins. It's that kind of play.
August Strindberg's Ghost Sonata is this combined with Nightmare Fuel. "She sucks all the gravy out of the food and replaces it with water."
The interrogation sequence from Anyone Can Whistle, an extended musical scene ironically titled "Simple." For elaboration, it's a thirteen-minute song that closes the first act, in which a dubious "doctor" has to sort through a crowd of people and determine who is sane and who is not. Eventually, he starts drilling the town council, and everything is in an uproar, until everything stops and he says "You are all mad." The lights go down on stage. When they come up, the cast is in seats, holding show programs, and they applaud the audience.
Philip Glass's opera Einstein On The Beach is all symbolism and no plot.
After Magritte doesn't make a lot of sense either. But that's kind of the point, as it's supposed to show how the characters in a surrealist painting (like, for example, a Magritte) got to the position they were painted in.
Harold Pinter's Old Times. A wife and husband (Kate and Deely) invite a guest (Anna) over for dinner and talk for the remainder of the story. The ending implies that Anna doesn't actually exist.
Gene Doucette's Deus Ex Quanta. A dark comedy about a murder investigation and quantum physics, performed in Anachronic Order.
Eugene Ionesco, like most absurdists, did this excellently. While some of his plays were actually strange and elaborate metaphors (like how everyone turning into Rhinocerouss in Rhinoceros was analogous to the support of Nazis within France), some were just weird.
His first play, the Bald Soprano. Inspired by the inanity of the phrases found within an English-French phrase book the play contains bizarre speech patterns, unusual repetition, faulty logic, a spontaneous and unpredictable clock, a married couple who don't know each other, and a rather overdone argument concerning doorbells. The play is hilarious, but it's always hard to tell which parts are purely humorous and which are symbolic. There is no bald soprano.
Hilariously Lampshaded (or possibly a Throw It In) in "Exit The King"— at a few points, the lighting goes strobe and the characters fling themselves about the stage to symbolize the change and upheaval going on in the play. After the third or so one of these, the King (played by Geoffery Rush in the most recent revival) yells "What the fuck was that?!"
A number of modern operas:
Marco Polo by Tan Dun — When you see something called "The Book of Timespace" and find that "Marco" and "Polo" are two separate personas, you know you're in for something. Critics went so far as to suggest that Tan collaborates with a prominent music critic (Paul Griffiths) to write the libretto, in order to immunize his work from criticism.
The Mask of Orpheus by Harrison Birtwistle — Gramophone Opera Good CD Guide advises listeners not to read the libretto the first time they encounter this work.
Cirque du Soleil has at least one show that qualifies as this. Mystere's Excuse Plot has two babies searching for their loveys in a strange world of animals, viruses, angels, demons, etc. According to the creators, this is all a metaphorical exploration of the origin and meaning — that is to say, the mystery — of life. Not surprisingly, it's not easy to decipher the symbolism without help from the website and/or the making-of documentary, but you don't have to do that to enjoy the show.
Tennessee Williams' Camino Real, or at least parts of it.
Pippin could qualify. For the first 90% of the show, the players are all aware it's a show — except for the actor playing Pippin, who's stated to be an actor but seems to think it's real. Then the script starts changing, characters die and come back to life, and Pippin is told to kill himself; he is saved by his love interest, which gets them banished from the troupe. Moreover, just before the show ends, the Leading Player says "Why, we're right inside your heads!" making you think it was All Just a Dream and question if anything really happened at all!