Continuity mistake: When Madame de Pompadour is walking in the gardens with her friend, the Doctor ducks down behind a pillar so he will not be seen. He puts his hands on the pillar but in the next shot they are gone, then they are back again.
Continuity mistake: When the two Cybermen meet the Dalek for the first time, they are exterminated. They fall directly in front of the Dalek, but when the camera angle changes to show the second Cyberman hitting the floor side on, you can see the Dalek in the background. The Cybermen both fell straight forward, so it shouldn't have been possible to see the Dalek when seeing the Cyberman side on in the next shot, as the Dalek is facing them both directly.
Continuity mistake: After Miss Evangelista is killed, we see her communicator as her "ghost" slowly fades. In one shot we see from a slight distance the lights left lit tick down to 3, then a short while passes, we cut back and the lights are back to 4. They then tick down to 3 again.
Add timeJon Sandys
Character mistake: In an early scene with Yvonne Hartman, she says "We've just measured the ghosts' energy at 5000 gigawatts." There are at least two mistakes here: (1) watts are a unit of power, not of energy (that would be joules); (2) later on, when Hartman is showing the Magnaclamp to the Doctor, she refers to imperial tons and says "Torchwood refuses to go metric", but a gigawatt is a metric unit.
Continuity mistake: In the scene when Donna asks the woman if she could borrow £10 because it's Christmas, you can see that Donna is wearing sports shoes instead of her usual white wedding shoes.
Character mistake: When the Doctor is taking Donna to 4.6 billion BC to watch the formation of the Earth, he says that it is the furthest back the TARDIS has ever been. But in the 5th Doctor episode "Castrovalva", the TARDIS drifted to 13.2 billion BC and the formation of the galaxy.
Plot hole: The Sycorax begin speaking English for all of the humans because the TARDIS is suddenly activated when the Doctor wakes up. However, it is stated elsewhere in the series that the TARDIS only translates for those who have been in the TARDIS, which Harriet and her translator haven't.