When Foster waves at the rescuers, the co-pilot looks to the side to see him. However the moon mobile has already turned away from him, and doesn't have side windows anyway.
When Colonel Foster visits Tina Duval and she is listening to the record player, before she answers the door she switches the player off, but on the soundtrack the record continues, just at reduced volume.
At the beginning of the events on the Moon, the Sun can be seen on the horizon. A few hours of story time later, it's high overhead, but the Moon is tidally locked to Earth, so the Sun's passage is 28 days, or 14 days of daylight. It would take a week before it was overhead.
Bill Grant continues to breath, even though 1) the Moonbase window has broken, and the room has depressurized; and 2) he's dead.
In one scene Cmdr. Straker is show to be driving on the right-hand side of the road whereas in an earlier scene Col. Foster is shown to be driving on the left-hand-side of the road.
The dust kicked up by the Moon-mobiles hangs in the air, showing that the action was filmed in a normal studio. On the airless Moon it would drop straight down.
Even though Foster's car is parked at SHADO HQ, he takes Straker's car to visit Tina.
When Paul Foster falls against a rock face on the Moon's surface, the entire rock face moves.