Continuity mistake: When Joey Zasa comes to Michael to complain about Vincent, Michael asks Al Neri to get him. Al leaves and we hear the door close behind him. Then Vincent comes, accompanied by his aunt, but Al is nowhere in the picture. But when Vincent does a Mike Tyson on Zasa's ear both Vincent's thug Squigilaro and Al come to separate the two opponents.
Continuity mistake: When Michael, Anthony and Mary are walking along a wall, Anthony is holding a sheet of paper which he is studying at the beginning of the first shot in the scene. The paper is the picture of Michael in his limousine that Anthony drew as a child (and is seen in Godfather Part II). However, Anthony isn't given this picture until later in the film. Notice that they're wearing the same clothes in both scenes. Clearly a much longer scene was cut into two parts, and the second half is seen first.
Other mistake: The close up of the Wall Street Journal mentioning Immobiliare seems to have operating instructions for a Windows program under the headline. The newspapers previous to this appearance also have a headline pasted onto another news topic altogether. The same happens later with the newspapers covering cleansing actions of the new pope. These papers date from March 27, 1980, although the events are supposed to take place in 1979.
Deliberate mistake: When the heads of the families assemble in the Palazzo Azzuro in Atlantic City and get attacked from a helicopter the rotors make the glasses on the table shake and the piles of fruit in the bowls collapse. As the Palazzo looks like quite a solid building this effect is a bit exaggerated.
Visible crew/equipment: In the scene where Joey Zaza is murdered, one of his henchmen gets gunned down with a shotgun. Just before the blast hits him, the side of one of the wooden street stalls magically changes to a breakaway so he can crash through it dramatically, instead of just slumping down against it as he would in reality.
Continuity mistake: When Calo assassinates Don Lucchesi, he grabs the Don's glasses and stabs him in the upper chest about three inches below the collar of his shirt, yet later, when Lucchesi is shown slumped in his chair and bleeding, the glasses are seen sticking out of his neck above the collar.