Continuity mistake: In the first "Godfather" film, we see Vito Corleone's tombstone, and it says there that he was born on April 29. In this movie, however, Sonny curses the Japanese for bombing Pearl Harbor on Vito's birthday. The attack on Pearl Harbor happened on December 7, 1941.
Continuity mistake: Shortly after assassins try to kill Michael at his home, he meets privately with Tom Hagen. As Tom enters the room, the table is empty. Then, after they are talking for a few moments, Michael offers Tom a drink from a bottle of courvoisier that has magically appeared.
Continuity mistake: In the scene where Roth's symbolic birthday cake is divided among his partners, we see a shot in which Michael is seen, in long-shot, putting a forkful of cake into his mouth while talking about the rebels. The thing is, it does not sound as if he is talking through a mouthful of cake. What is more, when the scene cuts to closeup, he is not chewing anything, even though it's just seconds later.
Continuity mistake: When Senator Geary talks to Michael in his Lake Tahoe office and the camera is on Michael there's an older couple (the woman is wearing a fur jacket) walking by outside. The camera cuts to Geary for a while, and when it cuts back to Michael the same couple is still walking by.