Continuity mistake: Magnus Greel puts Leela in the distillation chamber and switches it on. Cue red lights, special effects, and Leela writhing about. Then the Doctor arrives and throws a battle-axe into the works. Sparks fly, and the machine stops, presumably broken. Later, however, when Greel is pushed into the chamber, it turns on and does its thing just as if nothing had happened to it. Oh, and the axe thrown earlier disappears too.
Continuity mistake: In episode 1, when the Doctor and Leela leave the TARDIS, the door is open and is not moving. They then walk away from the TARDIS for a few seconds. When they hear the attack on Bullard (the Cab Driver) and run past the TARDIS again, the door is magically closed.
Factual error: When The Doctor questions the arrested Chinaman in the Police Station (in episode one), the Doctor claims to be able to speak several Chinese dialects. What the Doctor actually says to the Chinaman is "How Are You?" in Cantonese, but the rest of what the Doctor says is completely improvised gibberish that SOUNDS Chinese, but is, in fact, complete nonsense.
Factual error: The Doctor mentions at one point in the story that he had once been fishing in the River Fleet with the Venerable Bede. The Venerable Bede was a distinguished 7th century scholar, who lived his entire life at Jarrow, on Tyneside, and never at any time came to London.
Revealing mistake: At various points in the series, Mr. Sin is obviously a dummy and not an actor. But the most obvious is during the last fight when Sin jumps the Doctor. After a struggle, cut to a forward angle and the Doctor grabs and throws an obvious dummy to the ground.