Continuity mistake: When the Doctor and Ida descend into the core of Krop Tor, the screen charting their progress shows 39% Oxygen. After cutting away to an overhead shot of the two, the camera cuts back to the same descent display, only now the oxygen percentage (which has been descending constantly all this time) is at 42%.
Other mistake: When The Doctor tells the Atraxi "Oh, you're not the first lot to have come here, there have been so many," one of the monsters shown from their scan of Earth are a group of Ood. Even if in one of their uprisings they attacked the Earth, it would have been in the far future. The Doctor had no encounter with them on Earth, and he knows them to be peaceful, so why would they be amongst the aliens who have threatened the planet? The only Ood who has been to Earth at this point is Ood Sigma, and he didn't come to attack.
Continuity mistake: As the scarecrow-soldiers advance on the school to do battle, the scene cuts to Sister-Of-Mine, looking out from a school window. There is heavy rainfall outside. When we immediately cut back to the advancing scarecrows, it is a cold, clear night.
Factual error: Magnetic tape recorders weren't available outside of Germany during World War II. True the tape recorder could have been captured from the Germans, but, as it was "cutting edge" technology for 1941, it would hardly be likely to have ended up in a hospital office.
Revealing mistake: Dalek is escaping and enters the large room with the soldiers. He starts the sprinklers so he can kill them faster. When they show the dead soldiers they show some casings which were ejected from the soldiers' guns. The casings still have projectiles in them. If used, they would have been in magazines and would never fall out like they are shown.