Other mistake: When Miss Hartigan is testing the earpod control, the men start off facing her. She instructs them to turn to the right. They turn 90 degrees clockwise. She then tells them to turn to the left. They turn 180 degrees anticlockwise. However, they should have only turned 90 degrees anticlockwise, and end up facing her again. What they actually do is turn on the spot, as opposed to turning left. It takes a further instruction from Miss Hartigan for them to face her again.
Deliberate mistake: When the Doctor visits all his recent companions before he dies, he goes to a book signing of "The Journal of Impossible Things" written by Joan Redfern's great grandchild. When she is signing the book for the Doctor, you can hear the scratching effect from the pen but you can clearly see that the pen isn't touching the paper.
Plot hole: Davros said that the reality bomb would destroy all parallel universes. He fails in the Doctor's universe, but the Doctor said in an earlier episode that every decision creates a new parallel universe, like if Donna had walked out of the TARDIS with Jack Rosa and the Doctor. She wouldn't have imbued herself with Time Lordness, so Davros would have succeeded. This means that all the universes, including the Doctor's one, should have been destroyed, because if the Reality bomb only succeeds in one universe, it would destroy all the others as well.
Continuity mistake: At the end of the episode, Donna piles suitcases in the Doctor's arms in preparation to leave with him in the TARDIS, and places a striped hat box on top. The bags stay that way for a while as the camera cuts back and forth between them, and you can see the stack of luggage as the camera films the Doctor from behind. Then, as soon as the camera cuts back to the Doctor's face when Donna leaves to give her mother the car keys, suddenly he has one bag under his arm and another in his hand, and the rest of the bags are on the ground with the hat box stacked neatly on top.
Continuity mistake: After the Master changes all humans into copies of himself and Wilf says, "What have you done, you monster!?", it goes to all the different masters in the room saying, "Are you talking to me?" The Abigail Naysmith-Master is shorter than the Joshua Naysmith-Master. In another shot they are the same height. The Abigail Naysmith-Master is wearing very high heels and should be taller than the other Masters.
Deliberate mistake: Episode 4.14: "The Planet Of The Dead": At the beginning of the episode, the police are chasing the bus and they give the numberplate as starting with 'Whiskey zero four seven' (Or something similar). That is W047, yet later on when the flying bus comes through the rift in time/space, the numberplate is completely different. This is because whilst filming in Dubai, the original bus broke down and had to be replaced.
Plot hole: Near the end of the episode, the part-human Doctor is said to have all of the memories of the original Doctor. If this is true, then the part-human Doctor should remember the time the original Doctor spent with Rose, so shouldn't commit genocide or need Rose to help him to not be a man of killing.