Battlefield - S26-E1
Deliberate mistake: Look for the "creases" in the wall inside the TARDIS, which look as if the wall was just a sheet hung up behind the console. (In fact it actually WAS a sheet and not a "solid" wall - the TARDIS wall prop was apparently unusable due to water damage, due to poor storage between seasons. This also explains the lack of TARDIS interior scenes throughout the rest of season 26: it was not thought worth repairing the TARDIS set [or building a new one] when the show was in imminent danger of being cancelled after season 26).
Deliberate mistake: Tom Baker has some wounds on his lip (caused by actor Paul Seed's dog) which appear in episode one and four in the TARDIS scenes and disappear for the rest of the episode.
Destiny of the Daleks - S17-E1
Deliberate mistake: The shot of the alien spacecraft flying in the sky is the same in episode 4 as in episode 1, only it's reversed. The taking off and landing are the same too, except smoke and rubble have been added.
Deliberate mistake: One major mishap occurred just prior to editing, when a videotape containing various close-up and insert shots was accidentally wiped. At that point, there was no way that the director (Nicholas Mallett) could go back and rerecord the material, and so he was forced to work without it. This meant using footage that would usually have been discarded or replaced. The 'give aways' that reveal that Mallet had to work around the problems of the missing footage are (a) the lack of any close-ups of any of the actors in the final 10 minutes of episode 4, and (b) the continuity error involving the flames in the laboratory for the same period...the flames go from low to high and back to low from shot to shot.
Deliberate mistake: The shot of rocks falling in episode four is used twice - once when Unstoffe is shot, once after the captain fires the cannon.
Genesis of the Daleks - S12-E4
Deliberate mistake: Chief of Security Nyder wears an iron cross on his uniform which mysteriously disappears in episode 3 - apparently the producers felt it was too close a reference to Nazism so it was removed from the show. (Source: DVD commentary).
Deliberate mistake: In episode three, the Doctor and Vira meet a half-changed Noah. He holds a gun on them, then drops it at Vira's feet and the door is suddenly closed. This is because the scene was originally longer and involved Noah begging Vira to kill him before the Doctor shuts the door. It was trimmed down to get it past BBC censors.
Deliberate mistake: Benton is a Warrant Officer in this story, but the end credits refer to him as "Sergeant Benton".
Deliberate mistake: In episode 3 of 'The Web Planet' (titled 'Escape To Danger') watch as one of the Zarbi operators, his vision impaired by the costume, runs right into the camera. So far behind schedule was the recording of this episode, however, that this mistake was retained in the finished episode, as there was no time left for a re-take.
Deliberate mistake: The character of "Chicki" is played by a completely different actress in episode 4 to the one who played her in episode 1.
The Power of the Daleks - S4-E3
Deliberate mistake: During all the animated parts of "The Power of the Daleks" in numerous shots various illustrated character's such as The Doctor, Polly, and Ben have their buttons, pockets, patches, etc, flipped to the opposite side, sometimes even within the very same shot.
Deliberate mistake: When the Doctor and Benton are throwing fungus at the maggots in episode six, a shot of a chunk of fungus rolling down a hill between two maggots is repeated.
Castrovalva - S19-E1
Deliberate mistake: The newly regenerated Fifth Doctor is wearing completely different shoes than the Fourth Doctor did in the previous story. This is deliberate. The boots worn by the Fourth Doctor were actually privately owned by actor Tom Baker. Still, a mistake but this is why.
Chosen answer: The Master is insane. If he can't control the universe, he'd rather see it destroyed.
Captain Defenestrator