Revealing mistake: You can see the studio floor in the "white void" outside the TARDIS door in early shots during episode one of this story.
Other mistake: William has a bit of a nightmare, completely throwing the others during one scene by saying the same line ("It's not very likely") twice, and fumbling the line "You knocked both Susan and I unconscious". He also omits the scripted explanation for the melted clocks.
Continuity mistake: Susan's socks disappear during the opening credits.
Audio problem: During Hartnell's big speech about the creation of a solar system, a cough can be heard from off-stage.
Visible crew/equipment: When Susan runs out into the console room in a panic, the shadows of two crew can be seen on the wall.
Plot hole: The Doctor says he used the fast return switch to take them back to 1963; however, the last place the TARDIS had been before landing on Skaro was 100,000 BC.
Chosen answer: The Master is insane. If he can't control the universe, he'd rather see it destroyed.
Captain Defenestrator