Character mistake: In the scene where the dentist is doing a root canal, Bloat says "rubber dam and clamp installed?" and another fish replies "yes", but when the picture goes back to the dentist and patient you can plainly see that there is not a rubber dam on the patient. A rubber dam is a large piece of latex attached to a metal frame, to keep water from the dental handpiece from going into the patient's mouth. Although all of the dental jargon in the film is precisely accurate, this device would have been very noticeable.
Character mistake: Captain McCrea tells the passengers that it's the 700th anniversary of the Axiom's first flight. The Axiom has been in flight for 255,642 days. Actually, 700 years is 255,675 days. That figure includes the additional day in 175 leap years.
Suggested correction: Leap years only occur because of the earth's rotation around the sun. As the AXIOM is in space, there is no need to correct for the earth's rotation.
First, rotation is the spin of the Earth (which cause day and night). Revolution is the earth orbiting the sun (which causes years). However, this correction is not valid on the premise you're trying to present. Many films set in space still use Earth time, so a day is 24-hours, even though they're in space and there is no sunrise and sunset (although it's stated the Axiom operates on a 25-hour day). So they would use Earth's year, which takes 365.256 days. Since the Axiom isn't orbiting the Sun, it wouldn't experience a year, so they're using something else. The fact that they're slightly off suggests it's a writing mistake and there's no evidence they use an arbitrary 365.203 day year.
Character mistake: Mr Potato Head gets torn apart by baby Molly after the opening scene. When Andy returns to the room with Buzz everyone resumes their positions, but Mr Potato Head looks for his bits and sits under the shelf with the other toys.
Character mistake: When Mike pulls out the first Scare Report, the word "yield" is spelled wrong.