Revealing mistake: In episode 12, while the Vervoids are dying, look out for the ones wearing sneakers and tracksuit pants (complete with drawstrings)...
Audio problem: All of the scenes of the trial have some sort of sound problems. Among other things, sound bites from earlier on appear from seemingly nowhere, and, if you listen carefully, you can hear someone calling out the cue "Take 6", which should have been edited out quite easily, but wasn't.
Factual error: The Master states that Earth and its entire constellation were moved, but constellations are merely patterns of stars (as viewed from a given point, like Earth) that may be nowhere near each other - the Earth could not really be described as being in a constellation as any sort of specific entity that could be moved.
Revealing mistake: In the final episode, Glitz and the Master leave the console room while the Doctor is driven into submission, but they are quite obviously standing by the scanner screen.