Factual error: In the scene where all the heads of families meet shortly after Sonny's death, there is an exterior shot of a large grey-brick building with an American flag hanging from it. If you look carefully, the flag is a current flag with 50 stars, with alternating rows of indented stars. But, the scene is takes place circa late 1940's when there were only 48 states and all the rows of stars were even. (By the way, the flag inside the meeting room is the correct one.)
Continuity mistake: When Diane Keaton and Al Pacino have dinner, he gets up, puts on his coat and sits down again. Now watch Diane Keaton's glass: When the camera's on Al Pacino she's holding the stem, but when the camera's on her she's holding the body of the glass. Another mistake is that at the end of the scene the amount of wine in her glass suddenly doubles.
Other mistake: After Sonny is shot on the causeway, there are a very large number of bullet exit holes visible on the sides of the tollbooths, which could only have been made if the gunmen shot through the walls (not the windows) of the buildings from the inside. This is common in movies. In real life, bullet entry holes in wood appear as small black dots, not the more dramatic-looking splintered gashes made on the other side of the wood as the bullet exits.
Continuity mistake: When Tessio delivers the bulletproof vest and the fish, Sonny's hands are in one position (one below, one on top of the parcel), a fraction of a second later they are in a very different position (both on top, unwrapping). The parcel is half open while Tessio is carrying it but fully closed when he puts it on Sonny's lap.
Continuity mistake: When Don Vito Corleone gets shot, the camera angle from the side of the car shows Fredo bent over trying to pick up the gun he dropped, but when we see him from the camera angle from above, he's just walking to the front of the car. When the angle cuts back to the front of the car, Fredo is once again bent over trying to pick up the gun.