Continuity mistake: The registration plate on Ethan Hunt's bike changes between 'NI 89' and 'ND 69' on occasion. Clearest moments are NI 89 just as he goes into the flames on the burning bridge, and ND 69 when he's through them.
Continuity mistake: After Gerry gets smacked in the face by the 'broken' suspender clasp, we see the silver clasp land under the dresser. Yet, when he's moaning and lifting his leg to get onto the bed, we can actually see the clasp still intact on the front suspender that came loose, which is dangling right behind him (just look between his legs). Of course, in the next shot, that clasp is gone. Since the silver clasp is one of the plot points, this is bizarre to say the least.
Continuity mistake: When John and Catherine are in the hangar at the runway, the Cessna's tail number is N3035C. When the plane is shown in the air, the number is N3973F. When they land, the tail number has changed back to N3035C.
Continuity mistake: In the bedroom scene the girl is holding a clear cup full of beer. The camera goes off her and when it comes back she is holding a blue cup. The camera goes back off her then on her and the cup is clear again.
Continuity mistake: When Sam is in his bedroom, after he finds the Cube sliver it falls and dramatically burns a hole through the floor into the kitchen. But Sam's bedroom is over the living room (there is nothing on the ground floor below Sam's elevated bathroom, and in the first movie Jazz hides in this open space), and the kitchen is actually located between the den and dining room, at the other end of the house. If they were to follow the actual floor plan of the house, the Cube sliver would have landed in the area of the piano in the living room, but where would the fun have been in that?
Other mistake: When devising the plan to retrieve Watney, it is mentioned that the Hermes crew have had to lash together all the webbing on board to make the longest possible tether. When this tether is used, there is no evidence of any lashing together or other extensions or modifications to lengthen it. The tether is one continuous length and is stored on a reel that was designed for the length of tether gathered on it.
Continuity mistake: When Honey meets Hiro, she has earbuds in her ears playing music. She starts talking to Hiro, pulls them out of her ears, and they dangle out of her shirt. The dangling earbuds disappear in the close up shot of her and Hiro.
Continuity mistake: When Gordo leaves Simon and Robyn for a few minutes to handle a work thing, Simon starts mocking Gordo in his absence. Simon tells Robyn that Gordo wants to nail her and Robyn says "Stop it," but Simon continues his mocking of Gordo. At the end when Simon is listening to the recording that Gordo made of that conversation, Robyn's "Stop it" is absent.