Not to go off on a rant here, but…
Joseph Cotten’s revelatory, deceptively placid rant against the idle, rich widows he preys on in Shadow of a Doubt.
Cher’s rejecting Jack in The Witches of Eastwick – the one that ends, “…on top of that, YOU SMELL!”. 
Jack Nicholson’s dealing with a rude waitress in kind, in Five Easy Pieces.
Sally Kellerman storming into the camp’s C.O.'s tent and resigning her commission, in MAS*H.
Shirley MacLaine demanding opiates for her terminal daughter, in Terms of Endearment.
John Cusack explaining at length what he doesn’t want to do for a living, in Say Anything.
Ron Livingston explaining exactly what’s wrong with his job to the consultants, in Office Space. Of course, since he’s still under hypnosis and very mellow, this is more of a unique form of confession than an angry rant, but it’s still great.
Annette Benning “meeting cute” with the Prez, in An American President.
The alien’s “Stupid, stupid, stupid!” rant in Ed D. Wood’s Plan Nine From Outer Space. IIRC, that is. It’s difficult for the brain to continue to function when assaulted by a Wood opus, after all!
Peter Lorre’s “Stupid, stupid, stupid!” rant in The Maltese Falcon. (Since I just saw this on TCM, I can vouch for it.)
Monty Python & the Holy Grail: the Black Knight, Dennis the Communist, the Frenchman in the castle…
Monty Python’s The Life of Brian: the hermit breaking his silence; Brian telling the adulatory crowd to think independently.
Monty Python’s The Meaning of Life: The Grim Reaper going off on boorish Americans (“You Americans. You’re always saying things like, ‘I just want to say this,’ and ‘Let me tell you something.’…”) And I just want to say, as an American, that that was one fine rant, Mr. Reaper!
Babs’ defense of prostitution as a moral alternative to financially expedient marriage, in Nuts – “…I know women who sleep with husbands they can’t stand, just so that they can have a vacation home in the Hamptons!..”.
Lots of the dialogues in Heathers, which is appropriate, as it’s about teen angst amongst the mostly-rich, anyway.
And, of course, the treasure trove that is Withnail & I: Withnail Montague’s “We can’t go on like this…” cry of despair, his “I’m going to be FAMOUS!” phillipic, his “cake and fine wine” bit in the tea shop, and Peter Marwood’s “Get up, you bloody bastard…”.