Factual error: In the first close-up view of Mondas, the planet is spinning rapidly so that we can see that its continents are identical to Earth's. But this would mean that Mondas is spinning on its axis once every few seconds - tens of thousand of times faster than Earth. This would be such a fast rotation that the planet would not be able to hold together under the centrifugal force.
Continuity mistake: In episode 2, As Jo walks along the castle ledge to escape from the Ice Warriors' room, there is, at one point, a quick shot of a piece of the ledge breaking off under Jo's shoe and falling down the cliff face. However, the shoe isn't Jo's. If you look closely, you'll see that the shoe in that brief shot is more pointed and of a darker colour than the shoes Jo is wearing, and has a strap across the ankle instead of being a slip-on.
Deliberate mistake: In episode three, the Doctor and Vira meet a half-changed Noah. He holds a gun on them, then drops it at Vira's feet and the door is suddenly closed. This is because the scene was originally longer and involved Noah begging Vira to kill him before the Doctor shuts the door. It was trimmed down to get it past BBC censors.