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.
Other mistake: During the storm scene in the beginning of the movie, the astronauts' faces inside the helmets are brightly lit, meaning there's a light source pointed directly in their face. That's something that would render them mostly blind and unable to see and appears to be nothing but a dramatic effect for the camera. (00:05:00 - 00:08:00)
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.
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 Commander 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' 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' MMU be able to counteract the spin. The expected rotational speed-up effect is not seen.
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.
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.
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.
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.
Factual error: Towards the end of the film Watney is told to discard the heavy nose cone of the Martian Ascent Vehicle and replace it with a flimsy plastic sheet because the atmosphere is so thin that it will not be damaged despite the vehicle being accelerated to Martian escape velocity - nearly 14,000 kmh. But the storm caused massive damage earlier. It cannot work both ways - if the atmosphere is so thin that it won't dislodge a jury rigged plastic canopy from an accelerating spacecraft, it cannot possibly whip up a storm like the one we see.
Factual error: Prior to and during the orbital rendezvous rescue scene, the dialog states that a) the capsule with Watney is on a parabolic trajectory and will drop back to Mars b) the Hermes is on a swing-by trajectory that will carry it past mars c) the Hermes doesn't have the fuel to spare to do more than swing-by, i.e. They can't brake enough to even enter an orbit (yes, when one is on a fly-by, one has to brake to get into an orbit. Watney would be way slower still) d) the relative speed of the spaceships at the time of rendezvous is less than 20 m/s e) the spaceships are at a distance of less than 200m. All those statements can't be true at the same time. What orbit you are on is pretty much defined by your altitude and speed. If the speed and altitude of two spacecraft are the same, they are on the same (general shape of) orbit. Conversely, if they are on the same altitude but one is on a suborbital trajectory and one doing a swing-by, the latter is faster than the former by hundreds, if not thousands of m/s. By comparison, the rendezvous depicted in that scene would be like granny with her walker trying to "rendezvous" with an Amtrak train at full speed.The dialog states explicitly that the Hermes can't brake into orbit of Mars to pick up Watney because it doesn't have the necessary reaction mass to do that and get home too. If they can't even brake into orbit, they can brake to match Watney's speed even less. Even without that dialog, it wouldn't make sense either way. The acceleration required to do that would be measured in thousands of m/s. Translated into fuel, that would be dozens, if not hundreds of tons. No spacecraft this side of complete science fiction carries that much spare fuel, for the simple reason that lofting the extra weight to orbit would be prohibitively expensive.
Other mistake: Rich Purnell explains his plan to redirect the Hermes to Mars in order to rescue Watney, positioning people to represent planets and using a stapler to show the trajectory of the vessel. He is talking to experienced, qualified engineers and technologists working at a very high level on the space programme. They don't need drama school play acting to be understand things like this. He could have explained his plan in the most complex and abstruse terms and they would have been way ahead of him.
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 Vincent 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.
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.
Factual error: When Teddy is announcing Mark Watney's "death," the American flag is on the wrong side of the podium. This would be quite a goof for a government agency, especially one as image-conscious as NASA. Even in subsequent press conference scenes, the American flag is consistently to the left of the speaker, when it should be to the right.
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.
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.