When Barney trips over the menu and falls to the floor, his skateboard starts to roll away on it's wheels. In the very next shot the skateboard is now flipped over on its other side, now lying on the board.
When Grandfather gives Pistachio the "nest in a box", the shots change. One is the tables with the red tablecloth, which we see at the beginning. At a different angle, we see tables with tan tablecloths.
One of the precious items depicted in the film was the Apollo 11 lunar module, which landed the first humans on the moon. The bottom half of Apollo 11 (the Descent Module) is still sitting on the moon today - it was for descent only. The top half, the Ascent Module, was jettisoned into space before the astronauts descended back to earth. So it is impossible for the actual Apollo 11 to be stolen as depicted in the film.
When Pistachio and Jennifer go to the toy convention to find Bowman she says her name is Barbara, and Bowman invites her to his house. When Pistachio comes looking for her at the door (as the German Guy and the Snobby English Guy) he says he needs to talk to Jennifer Barker. Wouldn't he ask to see Barbara instead of Jennifer? Also Bowman knows her as Barbara and didn't say anything about him calling her Jennifer.
Two main revealing mistakes with the Turtle Man include: in between shots the glasses he is wearing change from being regular glasses to magnified glasses, and vice versa. And at one point when Jennifer is trying to get information from the cigar roller, you can see the bald cap Pistachio is wearing as the Turtle Man starts to come loose behind his right ear.