Deliberate mistake: When Commander Taggert and Lt. Madison 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 where she unzips her tunic to distract the aliens.
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.
Continuity mistake: When they are rolling the beryllium sphere back to the ship, Fred has his paper bag in his mouth in some shots and it disappears completely in others.
Continuity mistake: At the end, when the crew is bowing to the fans, Sigourney is shown moving her right leg back a small bit. However, when they show them standing up from the bow, her right leg is fully crossed behind her.
Revealing mistake: When Saris is torturing the Thermian commander, he stops and approaches Commander Taggart; while Saris is talking to Taggart you can see the lips and teeth of actor Robin Sachs behind the mask.
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: After Sarris' first missile has hit the ship, Jason runs towards Tommy and shouts, "Tommy, let's go, get us outta here, go go go go go." However, what we hear is not in sync with Jason's mouth until the last 'go'.
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].
Audio problem: When the aliens are talking to Taggart at his house, in a shot from behind the aliens looking at Taggart in the doorway, the audio is out of sync with the alien leader's mouth. You can see this when he says the word "destruction."
Audio problem: When Brandon's parents are listening to the news reporter talk about the crew not showing up to the convention, his words don't match his mouth at the end of the shot.
Audio problem: When they are rolling the beryllium sphere into the ship, Laredo is heard yelling, but his mouth isn't moving.