Mike Nichols required that Nathan Lane and Robin Williams filmed at least one good take of each scene sticking to the script before he would allow them to improvise (something both of the actors are known for).
Robin Williams's slip and fall during the shrimp scene (at around 1h 35 mins) was not planned. Williams really fell, and he, Hank Azaria, and Dan Futterman are holding back laughter. Williams can even be heard holding back laughter telling everyone to "Go" out of the kitchen.
Robin Williams was originally cast as Albert, but he wanted a change from flamboyant characters. Also, with his character, dressing in drag, the film would've basically been a repeat of William's earlier film Papá por siempre (1993). Thus, he asked to be cast as Armand, after Steve Martin, who was originally cast as Armand, dropped out. Mike Nichols agreed, and re-cast him. Martin later regretted this choice, as he never ended up making a film with Williams.
Hank Azaria created two different voices for the character of Agador Spartacus, one being somewhat of a more masculine voice and the other one being higher pitched. He was worried about the second one being too stereotypical until he asked a gay friend of his, who thought it was more realistic.