Deliberate mistake: When Commander Taggert (Tim Allen) and Lt. Madison (Sigourney Weaver) are about to press the blue button to stop the ship exploding, Lt. Madison's tunic is mostly zipped up. A few seconds later it is suddenly mostly unzipped. Those of you who have access to the DVD of the film can see that there was a deleted scene which was originally intended to go in here.
Continuity mistake: When Guy is introducing the actors at the start of the movie, while he is saying Jason's name, to the right of the screen you can see that Guy is holding a glass of water with lemon. The camera then cuts to an overhead shot of the whole crowd and Guy can be seen clapping with nothing in his hands.
Audio problem: When Gwen uses her Gilligan's Island analogy, Mathesar says, "Those poor people," and then Tommy says, "Oh brother." We are then shown Guy standing next to Gwen, and Guy says "We're screwed," although if you watch his mouth he actually says something like, "We're dead." This dub is probably for a similar reason to that of why Gwen and Tommy's swearing was dubbed over.
Audio problem: In the early backstage scenes from the convention, the other actors are angry at Tim Allen's character. The actor who plays Laredo clearly says "You are so full of sh*t", but it is edited to "full of it." Those two word edits ('s' word and 'f' word) were done to keep the PG rating, but the editing is sloppy. [This theory could also be expanded to the drug reference mentioned above].
Continuity mistake: When the 4 "nerds" first approach the captain to talk about their supposed problem the girl is standing in a different place than when they cut back. In a later scene the girl is in this position and I believe they reshot the earlier scene while they were doing the later scene.
Continuity mistake: During the opening sequence of Alexander storming out of the back rooms, you can see Tommy help keep him there. However within the next shot as they Call Gwen to the stage, you can see Tommy and the whole crew out on stage but Alexander and the Commander are still having the same dialogue.
Continuity mistake: Sarris' lieutenant Lathe is in two places at once during the siege of the Protector. He is seen in the atmosphere control room when the rock monster is beamed in and presumably dies later. A few moments later, he is on board Sarris' ship informing him that the crew has escaped custody.