Plot hole: After Romana and the Doctor hide in the relic room, the Captain enters, then bends down to inspect the door that conceals the Shrivenzale. Unfortunately, the actor turns to his right when he does so, which make it rather odd that he doesn't see the white-clad Time Lady standing right in front of him.
Continuity mistake: As the Master escapes from Axos, he kills the UNIT soldier standing guard outside, whose beret falls off as he slumps to the ground. Cut to the Master standing over the corpse, and the missing beret instantly replaces itself on the head of the corpse.
Add timeMatty W
Other mistake: In episode 4, as the Doctor, Tegan and Nyssa are being led away from the bridge, the Cyberleader loses his footing when he gets to the bottom step, and stumbles/trips up (He recovers himself very quickly; so quickly in fact, that this mistake was left in the finished episode).
Revealing mistake: When Fenric (having possessed the body of Judson) knocks over the chess pieces on the board, watch one piece (The King) in particular. You can see a 'hinge' made of white sticky tape attached to the board and the base of the chess piece, presumably there to stop the chess piece rolling away onto the floor.
Factual error: As the spaceship approaches the Earth of 65 million years ago, the Earth looks very much like the present day one. Hasn't anyone heard of "continental drift"? The Earth's land masses of 65 million years ago were not where they are now. (This has been fixed on the DVD)
Character mistake: The Doctor (Patrick Troughton) makes one the funniest fluffed lines in the entire history of Doctor Who the final episode. With all sincerity he says "We'll all be dead soon, unless you switch over to the SEXUAL air supply". (The word Troughton was probably looking for was "sectional").