Other mistake: When Miss Hartigan is testing the earpod control, the men start off facing her. She instructs them to turn to the right. They turn 90 degrees clockwise. She then tells them to turn to the left. They turn 180 degrees anticlockwise. However, they should have only turned 90 degrees anticlockwise, and end up facing her again. What they actually do is turn on the spot, as opposed to turning left. It takes a further instruction from Miss Hartigan for them to face her again.
Other mistake: The security guards shoot away the lock on the door, which they then kick down with no real effort. But the door would still be held in place by the hinges - they don't kick it with enough force to rip the hinges out, and even if they had, we can see that there's no damage on the hinge side of the door anyway to demonstrate their forcible removal.
Other mistake: Martha falls into the bog and Hath Peck rushes down to help her. She calls out "Help me, Peck!" but the subtitles read "Help me peg!" This is just one of many errors in this series' subtitles, which seem to have been added by listening to the dialogue without reference to the script.
Other mistake: When the Doctor and Donna get onto the roof, after he sonics the door shut behind them, as they run for the crane for the window washing cart he puts his sonic screwdriver in his mouth, holding it between his teeth, and pulls something out of his pocket. It looks like he might be carrying Miss Foster's sonic pen - except that at this point in the episode, it hasn't yet been revealed.
Other mistake: Shortly after everyone arrives on the Vinvocci ship, when the Vinvocci and the Doctor run through the door on their way to the flight deck, the Doctor comes back to get Wilf, who didn't initially follow because he was mesmerized by the sight of the Earth from orbit. It's obvious that David Tennant went just behind the door and paused briefly before coming back into the room, which is not what the Doctor is supposed to be doing.
Other mistake: Near the start, the Doctor and companion, just after having experienced a violent earthquake within the city of ancient Pompeii, discover someone sold the TARDIS after fleeing toward where they parked it and finding it gone. But where it had stood there is a floor to ceiling shelf of clay pots, very aligned, unshaken, unmoved from the shelves.
00:04:00James G. Mason