Continuity mistake: In the beginning where Indy is young, he fights some guys on that zoo train. When he's on the crate with the rhino in it, the background changes in each shot. Once there are trees behind and a split of a second later there are rocks, once there's a cliff and once a plain, and it all changes in splits of seconds as they change shots. (00:07:25)
Continuity mistake: At the airport, someone is reading a newspaper. Instead of the year 1938, the date of the newspaper is 1918 (a guess: I think this mistake happened because the art director wrote the date in a rush, stretching the "3" in a way it seemed a "1" when the people of "atrezzo" saw it).
Continuity mistake: When the tank is about to go over the cliff you can see that there is no one sticking out of it, but in the next shot, Vogel can be seen sticking out and yelling.
Continuity mistake: When the Nazi Colonel discovers Henry Jones sen. aboard the zeppelin, Indy asks him for his flight ticket. In the first shot, Indy is standing right beside the colonel (the camera is behind the colonel and Indy comes in from the right), in the subsequent shot, Indy is standing behind the colonel.