Continuity mistake: When Quaid uses the hologram, it's a mirror image of himself. However, before he enters the reactor room full of guards, we see him put the hologram device on his left wrist. His hologram (the one all the guards shoot at) has the device on his left wrist as well, instead of what would be the right.
Continuity mistake: When Quaid enters George/Kuato's private room in the rebel base you can see George's desk has a different arrangement from one shot to the other with Quaid's gun disappearing, a blue container swapped with a white on, the computer's screen having a cable plugged in or out and other objects moved around. (01:16:00)
Continuity mistake: In the scene where Quaid takes out Lori with a bullet to the head, she lands on the floor in such a manner that allows blood to run across the right side of her forehead. A few shots later Melina pushes Lori out of the way, repositioning the body on its opposite side. But once Richter discovers Lori moments later, blood is only visible running across the left side of her forehead. If anything, there should be blood running from either side of the bullet wound. (01:08:10 - 01:09:05)
Revealing mistake: In the last scene, when Douglas Quaid and Melina are holding each other and looking over the landscape of Mars while the sky gets colored, the camera starts from top behind them (looking down) then moves down while looking at them, so more and more of the Mars horizon gets visible. But the camera-tilt goes too far up, so at the top of the picture, you can see the blue-screen curtain and the pole and rings holding it. Douglas Quaid and Melina are just looking at a blue-screen. (This is visible on the video; on the DVD an extra black top-bar is covering it.) (01:48:55)
Continuity mistake: When Doug and Melina are being chased by Richter they leap from the catwalk in the big dome. When they reach the other side, from one camera angle, they grab the horizontal bar on the dome scaffolding. From the next angle they are holding onto the diagonally vertical bars swinging inward. (01:06:30)
Other mistake: When Quaid goes through the x-ray machine the first time, the person behind the man with the dog isn't shown to be carrying anything (there's no purple object shown). But then later the purple object is seen.
Continuity mistake: In the apartment, Lori grabs a knife, and after the first slice, switches the knife to her right hand. After the second slice, the knife can be seen midair as she tosses it from left to right again.
Factual error: Martian sky is very inaccurate in the movie. It is depicted as much darker and redder than it would be in real life (Mars' sky is a butterscotch color during daytime, bluish at sunrise/sunset and far brighter than in the movie). Also, both of Mars' moons are very small, asteroid-like bodies and would appear as tinny star-like points of light to an observer on the surface of the planet.
Other mistake: After Arnold kills Harry and his gang at the beginning, how did that blood splatter get on the wall behind him to his right? He shot the henchman standing in that position from the back and the other who was shot was shot standing to the left side of the wall. So there is no explanation why the splatter is there.
Continuity mistake: Arnie has come back to the apartment after shooting his coworkers and asks Lori if this is a hallucination. He then reveals fresh blood smeared all over the palms of his hands but he had grabbed her before he asks the question. Lori is sitting in front of him without a drop of blood on her.
Deliberate mistake: While Cohagen is on Mars, he calls Richter and chides him for trying to kill Quade. Richter is in a car on Earth, yet they're having a real-time conversation, which is impossible because of the distance. [While this is true, a movie with 4 to 21 minute pauses between lines would take rather a long time].