Audio problem: When the Doctor arrives at UNIT HQ in his stolen car, he speaks to a guard. We see close ups of the guard twice...but watch the brief clip of The Doctor (Jon Pertwee) between the two and you can just see that his lips aren't moving even remotely in synch with the words. (In fact, it is just a repeat of the previous shot of The Doctor, with new lines dubbed in).
Audio problem: In episode 3, when the dog is barking at Sam's house and is abruptly silenced (apparently by an Auton), the sound is obviously not a dog barking. It sounds a bit like a man pretending to be a dog, and going "woof woof" not very convincingly.
Revealing: When the Doctor falls out of the TARDIS, near the beginning, he nearly manages to pull off the entire TARDIS door.
Audio problem: The music accompanying the closing credits fades down part way through (at a different point in each of the four episodes) and simultaneously fades up at a different point, the net result being a rather disjointed-sounding edit.
Plot hole: The Brigadier met the Doctor previously in 'The Web of Fear' and 'The Invasion', both Second Doctor (Patrick Troughton) stories. In 'The Web of Fear', the TARDIS was trapped in the tunnels, not to be seen until the end of the story...and in 'The Invasion' the TARDIS was invisible. So how does the Brigdier know that the Police Box is anything but an emergency phone booth?
Continuity: Taltalian's French accent mysteriously gives way to an English one in some shots, most notably on location, e.g. when he catches an escaping Liz and says "Get in Miss Shaw".
Continuity: There are a handful of similar continuity errors, but one that stands out in particular occurs at the end of Episode 1. The Doctor and Liz go into the lab and Taltalian steps out from behind the door wearing glasses and brandishing a gun. However, in the recap at the start of episode 2 he isn't wearing glasses and moves differently.
Plot hole: Just after the Doctor meets the Brigade Leader, he looks at the countdown and says in a very loud voice, "only 3 hours and 22 minutes". Several minutes later, when he leaves the office after stunning Benton, the countdown still reads 3 hours and 22 minutes. Did the countdown clock stop?
Visible crew/equipment: When Liz and the Brigadier are arguing toward the end, a shadow moves in the background. It's in the wrong place and too soon to be the Doctor returning.
Continuity: Throughout the serial, Bessie's tires are dry in studio shots (inside the Doctor's shed) and wet on location shots.
Revealing: When Stahlman is killed - the second time - the shot of his green primord face reveals the actors unpainted neck under his beard.
Continuity: Liz's hairstyle is different in indoor and outdoor shots.
Other: The power output in the Doctor's shed read "Megga Volts" instead of "megavolts"
Revealing: In episode one, a wall wobbles as Benton hammers a nail into it.
Visible crew/equipment: In episode two, the shadow of the boom mic crosses Jon Pertwee's head as he's giving the "epsilon coordinates" to Liz.
Revealing: In episode 6, when the Brigade Leader shoots the Primord Stahlmann, he falls down but no bullet holes are made in his costume. Thanks to Producer Barry Letts for spotting this one on the DVD commentary.
Continuity: In episode 4, the Doctor's bow shrinks from when he talks to Liz Shaw while in prison, to when he stows aboard a jeep later. In the DVD commentary, producer Barry Letts blames himself for this mistake.
Revealing: In episode 5, when the Doctor kills the Primord with the fire extinguisher, he says 'he's dead this time' but the Primord is breathing.
Revealing: In episode 6, during the generally poor fight sequence between the Brigade Leader and Greg Sutton, Greg's first punch to the face misses the Brigade Leader by miles.
Revealing: In episode 3, when the Doctor is squaring off against the drooling madman, the madman is inside the perimeter railings. When the soldiers shoot him and he 'falls', however, he has moved outside the railings, enabling the stuntman to jump safely.