Revealing mistake: The Cyberman that falls from the roof in the final episode is obviously an empty costume. (This is another example of a mistake being repeated from an earlier episode. See the earlier story "The Tomb of the Cybermen" for a note about the same 'revealing' mistake happening).
Factual error: How did potatoes come to appear in a 12th century kitchen? The potato was unknown in England until Sir Walter Raleigh brought them from back from the Americas on one of his voyages in the late 16th century, at least 400 years after the period in which this story is set.
Continuity mistake: The rhyme the Doctor recites in episode 3 is different from the original rhyme heard in episode 1. (Apparently, the Doctor [William Hartnell] forgot his lines in episode 3, when he came to the point of reciting the rhyme. So he just improvised what he thought they were, rather than what had been scripted).
Plot hole: It is never explained why so much of the story takes place in England. All of the equipment for Sutekh's rocket comes from his tomb and could just as easily - probably more easily - have been set up right there in Egypt. And at the end, instead of just walking out of his tomb, he takes the spatial corridor to England just so he can get caught in the Doctor's trap.
Continuity mistake: When the Doctor and Peri first land on Androzani, mountains are seen in the background. They disappear when Peri picks up the silica. A couple of shots later, they're back. (This mistake was mentioned on the DVD commentary; the mountain effect was left out for financial reasons).
Plot hole: The freighter traveling back in time makes no sense. According to Adric, the Cybermen's computer is responsible, yet the Cybermen possess no such technology; their machine's task was simply to navigate the freighter towards Earth. The idea that time travel could happen by accident is also absurd.
Factual error: Giuliano explains to Sarah that he thinks the world is round as though this were radical. But this was the common belief throughout the late Middle Ages. The argument was about how big the Earth was and the ignorance about the existence of a continent between Europe and Asia.