Rod Steiger vs. Marlon Brando
As part of his contract in On the Waterfront (1954), Brando only worked till 4:00 every day and then he would leave to go see his analyst. His mother had just died and he was having trouble dealing with it.
During the famous "Brother Charlie" in the back of the cab, Brando left the set early, leaving Steiger to deliver his lines to an empty wall. Steiger carried a grudge about this until 1997, when the two finally made peace.