When Richter and the soldiers ambush Quaid in the reactor, you can see one soldier firing a shotgun over Richter's shoulder in the shot of him firing and grinning. In the next shot of him, we see the soldier getting into position and shooting over his shoulder.
When Richter and Helm are waiting by the hotel's service elevator for Lori to bring Quaid down and realize something is wrong, they run over to a normal elevator and go up, yet they end up coming out the service elevator on the 6th floor.
During the escalator shootout, Quaid uses a bystander's body as a bullet shield and the body is shot to pieces; Quaid then throws the very bloody body onto Richter and his henchmen to slow them down. Moments later, as Richter and his henchmen chase Quaid into the train station, it is obvious that they haven't a drop of blood on them, despite the fact that the bloody corpse landed right on top of them.
During the cab chase through the tunnels on Mars, Quaid shoots Richter's gun from his hand. Richter then reaches down and picks up a large shotgun. However, he wasn't carrying this weapon when he entered the cab and it's highly unlikely he just found it on the seat.
When Quade first meets Melina at The Last Resort, she takes him upstairs and after she closes the door, she smacks him across the face hard. Her pendant is jostled, but in the next shot just before she hugs him, the pendant is stuck in her hair.
After the fight at the construction site, Quaid rushes home. When he is turning off the lights, he flicks off individual switches, yet while fighting with Lori, he flicks on one light switch to see who he's fighting, and the room becomes completely lit.
The two moons of Mars appear in several background shots, but they are incorrectly depicted. They are both shown as being much larger than they would appear in real life - Phobos would appear to be about a third of the size of the moon, and Deimos as little larger than a star. In addition, Phobos revolves around Mars faster than Mars rotates, so it would appear to rise in the west and set in the east, and should have moved visibly with respect to Deimos even from one shot to the next. The two moons are always shown in the same relative positions, however.
The exterior shots of Mars depicts the Martian sky as having either a dark orange or deep crimson color to it. In reality, the Martian sky is not as deeply colored, and actually a light pink or light orange color at times. Furthermore, the night sky on Mars is shown to be just a slightly darker red/orange than the day sky, when in reality the Martian night sky is just as black/starry as it appears on Earth.
When Richter and his crew are waiting to ambush Quaid at the reactor, they shoot at him from opposite sides not knowing that it's Quaid's hologram. This means that the bullets are passing through air (unless it's a bullet-absorbing hologram). However, none of the shooters facing each other gets wounded, nor killed.
At several points in the film, characters either shoot at the glass in the Martian domes and crack it, or are advised against doing so. It seems odd that this futuristic, space-faring culture would not install bulletproof glass in these domes, given the dire results of the windows being destroyed. Throughout the movie, several references are made to Cohaagen constructing Mars Colony by cutting corners, building cheap domes, and pocketing the cash for himself. Also the cheaply-built domes not adequately filtering out the solar radiation explains the mutant population on the planet.
When Quaid is running through the X-Ray hall, he's surrounded on one side by the guards and the other side by Cohagen's thugs. He then crashes through the glass and the guards are no longer there. Missing an obviously easy capture.
In addition to the fact that some of Richter's men would have shot each other when they're firing at the Quaid hologram (since they're position in a way that the trajectory of their shots would impact each other as they pass through the hologram's nothingness), none of the troops or Richter seem to notice during the 15 or so seconds of shooting that there are no impact wounds or blood from the high-caliber weapons "hitting" Quaid. He does fake that he's being gravely wounded and then shot to death but there's no way he could fake having dozens of gunshot wounds, especially since it's clear he isn't wearing any protective body armor or anything that would otherwise explain him getting shot with no visible damage.
The security guards wear some kind of portable computer (display and keyboard) on their wrist. During the fight at customs, a close-up reveals that it's just a Sharp pocket calculator taped to a glove.
When Richter is shooting at Quaid from the top of the escalator he uses a human shield then turns and throws the body at the goons on the escalator below him you can see the stuntman's eyes open and face cringe as he flies through the air.
When Quaid is in the slum hotel room, the phone in the room starts to ring. The person on the other line is a 'buddy' of his from the agency on Mars who says that he was suppose to find him Doug if he ever disappears. After Doug wraps a towel around his head to muffle the bug signal, the friend tells him to go to the window to see a suitcase that was going to leave for Quaid. The 'friend' on the visual phone looks up to the right as though he's looking up to Quaid in the hotel room. In fact, Quaid's 'friend' has to turn to the left and look up, not right as shown on the videophone.
After Quaid arrives on Mars for the first time and the outer dome is shattered and people are hanging horizontal from the supposed "pressure" release, the customs clerk who hits the button to seal the "airlock" is hanging on to a platform that is not anchored down, and wobbles as he grabs onto it.
The goof items below may give away important plot points.
During the fight at "The Last Resort", Thumbelina stabs
Helm in the stomach. When the camera pans out, the knife and blood are coming out of his chest (originally, Helm was supposed to be gutted from crotch to sternum, but was edited out for the R-Rating).
When Quaid and Milena are suffocating and decompressing on the surface, the atmosphere cloud hits them. It can clearly be seen that, as the air hits them, they appear quite normal. However, in the next shot, they are swollen again and their eyes are bulging.
Just as Quaid and Melina are about the be crushed by Benny's ore driller, the machines drill bits rips into Quaid's right arm and Melina's left arm. Afterwards, for the rest of the movie, though their shirts are shredded, their arms are unbloodied.
After Lori has been shot dead, her head is leaning to the left and the blood has started seeping. Melina flips her over to the other side, so her head is leaning to the right. When Richter turns her head forwards the blood is seeping from the right, with no blood stain on the opposite side.
The actual "ballooning" of the body while exposed to depressurization and lack of air won't take place on Mars as the planet itself has enough pressurization to prevent any "ballooning" of the human body.
During the final scene, the alien reactor begins pumping out oxygen, providing a stable atmosphere within minutes. It would take far longer to pump out enough oxygen to cover the planet, and for as long as that is happening, the entire area around the reactor would be subject to hurricane-force winds.
The gravity on Mars is 0.38 of Earth's gravity, but in all Mars scenes people are moving about as if they were in an Earth-gravity environment. Within buildings, this could simply be the result of artificial gravity created by future technology, but as Cohaagen, Quaid and Melina are ejected onto the surface of Mars, it is clear that they fall and land as they would on Earth.
All of the equipment that Quaid uses on Mars (the portable hologram generator, the Martian money, the Brubaker ID, etc) was all contained in the briefcase that Hauser's buddy left for him on Earth. How did Quaid acquire the Fat Lady mask? It is clearly too large to be stored in the briefcase (especially with the laptop device taking up most of the room) and you can see that the green dress used for the Fat Lady costume was also in the briefcase (look carefully when Quaid pulls out the laptop device and you can see part of the dress come out of the briefcase).
Towards the end when Quaid is using his holographic watch to fool the guards who have surrounded him on the catwalk, they begin shooting through his "holographic" image but no-one gets hit on the other side, even though they are only a few feet away.
During the reactor elevator scene when Richter's arms are disembodied and he falls down, Quaid holds the remaining arms and says "See you at the party Richter!". When he throws the arms off the elevator in the far shot, you see the arms fall behind the platform of the elevator, but however you do not see them fall past the platform and instead just disappear.