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.
Revealing mistake: In episode 2, when Scaroth locks the Doctor, Romana and Duggan in the basement dungeon, and Duggan lights the lamp, notice that Tom Baker is standing between that lamp and the actual stage light used to brighten the scene. So, as the lamp is lit in front of him, Tom Baker's BACK (and not his front) glows bright under the increased illumination.
Continuity mistake: When the Count's henchmen come into the café to get the Doctor, Romana and Duggan, they usher them all out at gunpoint. When you see them in the location filming in Paris, Duggan's not there...but back at the Count's place (in the studio in London) he's reappeared.
Plot hole: The Atlantic Ocean didn't exist in Early Devonian times (c.400 million years ago), so Scaroth's ship could not have been where the Doctor claimed. The Atlantic was formed when Europe and Africa separated from North America around 160 million years ago and is still growing. What is now the sea-bed was once covered up by several miles of Continental Shelf. The Early Devonian landscape would have been far from barren as plantlife was well established by then. So, either the Doctor's theory that Scaroth's exploding ship caused the creation of life on Earth is wrong, or it was much earlier than he said.