Air hoses on the spacesuit helmets switch sides during the spacewalk.
During the fight scene between the saboteur and Tony at the supply outpost, their oxygen tanks change back and forth between realistic ones (cylindrical) and fake painted boxes between shots.
The rocket shown on the launch pad and during the early part of the flight is vastly different from the one shown later in the flight and for the moon landing. Initially, it's mostly white with a blunt nose, multiple engines and no fins. Later, it dark-colored, with a pointed nose cone, a single engine and large fins.
When the men go outside the spaceship to perform repairs, the colors of their space suits change.
Fires are shown burning on the moon. There is no atmosphere to sustain flames.
A string can be seen that lowers the space ship as it is landing on the moon.
Astronauts hear and react to objects falling on the moon. There is no atmosphere to conduct the sound.
When the fuel depot explodes, here is a huge noise. No noise could be heard on the Moon since there is no atmosphere to carry the sound.
The space suits are clearly not pressurized, as they wrinkle as the characters move and hang loosely on the body. Also, the air supply hoses lack a basic safety feature, as the necessary check valves to prevent the suit from depressurizing when the air hose is disconnected aren't present. Finally, the characters move as they would on earth, not in the one-sixth gravitational field on the moon.
Once the ship had attained escape velocity the engines would have been shut down. Even so, the external shots show them as still be under full thrust burning fuel that will be needed for landing. This also goes against Doug's statement that the ship was rotating thus providing an artificial gravity of one g. Under full thrust, they would be experiencing multiple g's.
As Dr. Brandon crawls between the rings of the Time Tunnel guards run past behind him and as Dr Brandon is standing up, two of the guards can be seen standing through the next set of rings in the top left corner of the screen and one even rests his hand on one of the rings as the camera pulls back.
In the Time Tunnel lab, Beard watches himself ten years into the future threatening Doug and Tony because he doesn't believe their story, yet no one bothers to ask why he doesn't believe them when he was inside the Time Tunnel ten years prior.
Toward the end, the fuel canisters of extra fuel needed for them to get off the Moon are not very big (as Doug says they weigh 100 lb. each). Four of those canisters can't possibly be enough fuel to launch the rocket. In fact, the shed wasn't big enough to hold a sufficient amount for their purpose.
There would be absolutely no need for astronauts to carry loaded pistols to Mars.