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?
Revealing mistake: In order to save money filming the lift-off of Crayford's rocket from Oseidon, the producers decided to use stock footage of a Saturn 5 blast-off. Unfortunately, that meant the words "UNITED STATES" in large red lettering are seen on the side of the rocket...which is a little bit of a giveaway, especially as the rocket is supposed to be a BRITISH space probe.
Plot hole: When the airlock door is opening at the end of the final episode, the Cyberman is the only one who appears to be affected by the air rushing out. Although 'rushing out' is probably overstating the speed of the 'explosive decompression'. It's not so much an explosion as a gentle breeze...
Plot hole: In episode 4, watch what Romana does to Professor Kerensky's time machine in order to make it work. It is surprising that the time machine works at all, considering she is supposed to be in Paris at the time. Romana wires up the time machine to a British (i.e. 3-pin 240 volts) electric plug instead of a "continental" 3-pin 110 volt plug.
Continuity mistake: In episode 1, when the Doctor opens the garage doors with his 'door handle', note the two men in the background, the position of the car behind him and the white pillar. In the next shot, the men and pillar disappear and the car moves to a different spot. In the shot after that, the tanker truck is now parked in front of a pillar. You can see near the beginning of episode 3 that there are two entirely different buildings that the tanker had been filmed in front of.
Continuity mistake: 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.