Factual error: After Watney patches the blow out of one of the HAB's airlocks with plastic sheeting, tie down straps, and duct tape, he pressurizes the HAB and the plastic sheeting pushes out like an inflated balloon. Assuming the plastic and duct tape would hold this is correct, however the plastic would be much more taut given the pressure difference inside and outside. The real mistake is later in the scene during a sand storm the plastic flops in and out. The plastic would remain tautly inflated, since the inside pressure is much greater than outside, and since the HAB is airtight the storm would have no equalizing effect to cause the plastic to be sucked inward.
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.
Factual error: Lewis replaces Beck on the EVA to rescue Watney. On a NASA mission, each crewman is a specialist in several areas. While all of the crew have trained on EVA, Beck is the specialist for Ares III meaning he practiced EVA protocol and maneuvers 2-3 times as much as any other crew member. So while it is a nice dramatic moment for Lewis to replace him, a real mission commander would trust the best trained personnel to do their jobs, as she is actually lowering the chances of success by replacing Beck.
Factual error: When Captain Lewis in the tethered MMU jet-pack-chair is catching Mark Watney flying Iron-Man style via air jets from the hole in his glove, Mark misses Lewis's hand and grabs the tether, with some angular velocity relative to Lewis. As Lewis pulls the tether hand-over-hand to bring Mark closer, the angular momentum should spin them faster and faster like ice skaters pulling their arms in or Argentinian bolas (weighted balls connected by a cord) winding faster and faster around a target. Only when solid contact is made would Lewis's MMU be able to counteract the spin. The expected rotational speed-up effect is not seen.
Factual error: In the scene near the end where Mark is blasting off from Mars and before second stage engine cut-off, loose bolts are floating around the MAV cabin as if in zero G. With the engines still firing the acceleration would be pinning them to the deck. Even after SECO, it would take some time for the bolts (and probably a lot of Mars dust too) to jostle away from the floor and distribute evenly and randomly through the volume of the cabin.
Continuity mistake: When Bruce Ng is video calling Teddy Sanderson and Mitch Henderson to explain the modifications Watney will need to make to the Ares 4 MAV, there is a point in which the model MAV behind Bruce contains only a chair, all other chairs and the control console having been removed. Then, as Bruce and Mitch explain to Teddy that since Martinez will be remote-piloting the MAV, Watney doesn't need the controls, there is a shot of Bruce pulling several chairs and the control console out of the model.
Character mistake: In a scene when Watney is recording his video log, he imagines what would happen if his pressurized shelter suddenly would break. He concludes that he would implode, but the correct wording would be explode.
Continuity mistake: When Watney is putting duct tape on the plastic patch over the hole where the Hab airlock blew out, the first shot shows him outside the Hab putting the last strip in an evenly distributed set, from about the 10 o'clock position down to about the 4 o'clock position. Then (after a quick reverse shot from the interior), suddenly he is placing a new strip up from the 8 to 2 position, covering a spot that was already covered when he did the 10 to 4 strip, but which is now suddenly blank.
Continuity mistake: During the montage where Watney is preparing the potatoes, there is a shot outside in his suit where he attaches a cable or hose to the MAV outside the habitat. You can see the large engines of the MAV above Watney's head and the letters "MAV" on the leg of the vehicle, as well as the remains of the wrecked Rover 1 next to it, right where Martinez left it. By this point in the story, the MAV has already departed the Ares III site. This is supposed to be the MDV - the Mars descent vehicle - and Watney refers to the MDV when he discusses where to get the hydrazine - but it is the bottom of the MAV that is shown.
Continuity mistake: In the shot where Matt Damon finally gets captured by his rescuer, he is face to face with the woman astronaut and when he smiles his upper teeth are Hollywood white. In earlier shots, his teeth were shown stained and nasty. Also, shots soon after the rescue "white upper teeth" shot again reveal stained teeth.
Deliberate mistake: The atmosphere on Mars is only 1% as dense of that as Earth, so 175kph windstorms would feel like a light breeze. They would have very little effect on the astronauts or MAV. The writers of the book and the film were aware of this, it was a small cheat to let the rest of the story unfold.
Factual error: When the crew is walking around the area of the Hermes with artificial gravity created by spinning the ship, the angle of the crew members' bodies should be perpendicular to the curved floor of the ship. However, in multiple shots, the crew standing at different locations of the room are seen standing at the same angle, directly upright.
Continuity mistake: When Watney decides to retrieve Pathfinder, the date is sol 79. On earth Vincent says Watney has been traveling in the same direction for 13 days, and maps Watney's position as about one quarter of the way to his target. However Watney's helmet readout shows him arriving on Sol 94, only 14 days after he probably left the hab on sol 80.
Other mistake: Near the end when Matt is awake after launching the MAV, Commander Lewis mistakenly calls Mark Watley "Matt" when telling him the plans.
Continuity mistake: This mistake is in the extended edition: When Watney decides "I'm not going to die here", he packs up his five crew mates' belongings in plastic boxes and stows them in one of the two bottom bunk beds, from which the mattress has been removed. Then Watney is shown struggling out of bed, presumably the next morning, and a wide shot shows all six bunks. Every bed has a mattress and bedding on it and there are no boxes in sight.
Other mistake: Upon arriving at the MAV, when Watney opens the door to exit the rover to head to the MAV, the home-made balloon on top of the rover does not deflate, as it should when interior pressure of the Rover is absent.
Other mistake: When Vincent Kapoor first goes to JPL in Pasadena, he gets out of the car and the letters "JPL" are shown on the door. Since the shot is from the inside looking out at Kapoor, the letters should be backwards. No one labels a building for viewing from the inside.
Other mistake: When Watney is in zero-g there are loose bolts floating around him. There is no way he would have left loose pieces in the MAV's cabin. Things left unsecured in microgravity tend to float around and damage delicate electronics. If something is not being used it's lashed down or secured in a cabinet. He has one chance to to get rescued and he wouldn't risk dying after over a year of surviving on Mars on something so foolish as unsecured equipment in the MAV cabin. He would have cleaned out every last bit of loose gear before launch.