It's Too Dark!
Obviously, Paramount didn't like a lot of the choices that were made for this film (which all ended up being iconic choices, by the way.) During post-production, the studio exec's apparently hated the fact that the film was so dark. Coppola and Willis had to convince them that it was accurate to the film (which it was.)
Wired Shut
Michael Corleone had to have his mouth wired shut in the film. So, Al Pacino decided to actually have his mouth wired shut for those scenes! Now THAT is some impressive method acting.