Audio problem: When Bond is being shot at while motor-boating away from his pursuers in Venice, at one point during the chase, they shoot right next to a rope on his boat and a sound from a silenced handgun is heard when the bullet hits the wood. No one is using a silenced handgun.
Continuity mistake: When the bad guy is struggling with Bond at the beginning trying to get the parachute, just before Bond kicks the bad guy away for good, there are shots of Bond's face and he's wearing the kind of eye protection goggles Jaws has on, only in a clear color. Bond didn't have these with him and never put any on, and in shots after the bad guy gets the boot, he doesn't have the goggles on anymore.
Plot hole: Near the end of the film Drax tells Bond he had his own shuttle hijacked because he needed it after another shuttle malfunctioned. OK then why didn't Drax just contact whoever he needed to before the Moonraker left on the 747 and have the working shuttle returned to his base before it left. He obviously had the time to get two of his people hidden in the shuttle so a few phone calls would have fixed his problem and the British government would have never had to involve Bond at all.
Factual error: Vandenburg does not have the capability to launch space shuttles and even if it did you can't just launch a shuttle just like that. It takes weeks if not months to prepare a shuttle for launching, the booster rockets have to be properly cooled, etc. and somehow the Marines conveniently have a shuttle ready to go on a whim.
Other mistake: Bond is pushed from the plane without a parachute, but can see he wearing a altimeter on his wrist.
Factual error: When the space station starts to rotate to generate artificial gravity, the crews from the space shuttles can be seen walking along the tunnels. The gravity would be pulling them back towards the shuttles - acting outwards from the center of the space station.
Revealing mistake: When Bond and Chang are fighting in the glass museum, Bond cuts Chang's stick in half with his foil. If you look carefully, you can see a mark on Chang's stick where it is to be cut.
Continuity mistake: When Drax is jetisened from the Space Station by Bond, he flips end over end but at some point in the last moments we see him his body moves upward as though he hit some turbulence and then it moves downward. This isn't possible in space because unlike skydiving where one can manipulate his body in flight, in space one can't change his momentum without something to counteract the direction one's headed in like jet propulsion or running into an object that would allow them to stop the way they were previously headed. Drax has neither the jet propulsion since he is ejected in just his suit and doesn't run into anything which would change his direction.
Plot hole: The British government sends its best agent to the headquarters responsible for a missing shuttle, and Drax immediately wants to kill the man sent to talk to him. Surely Drax would have been smart enough to know if he had Bond killed both the American and British governments would have investigated him, even if he could chalk up Bond's death as an accident.
Plot hole: It would be impossible for Drax to decontaminate the laboratory and remove the equipment in Venice and return it to the room we see in the amount of time given.
Continuity mistake: When Bond shoots the man in the tree, he is wearing a hat. He is still wearing it as he falls, but when he is shown on the ground, it is nowhere to be seen.
Revealing mistake: When Drax is about to destroy the American shuttle, Bond, at the last minute leaks out all the gravity and everyone starts floating. In the next shot or two when you see Bond starting to float, a wire is visible that is used to make him hover.