A New World - S3-E20
Factual error: Connor has been raised since he was a few months old by Holtz, a man with a British accent, in a dimension where there are no other humans. There is no possible way he could have an American accent, yet from the moment of his return he speaks with one.
Suggested correction: The demons in Pylea spoke English with American accents (possibly due to Wolfram and Hart's interference), and in tie-in comics, demons in the dimension Connor and Holtz went to were shown speaking regular English, though these comics were released years later.
Sandy's Rocket / Squeaky Boots - S1-E8
Factual error: Sandy's dome is both airtight and watertight to keep any water out of the dome and any air inside the dome. When the rocket, positioned outside of the dome, begins to launch, smoke is coming inside the dome. This shouldn't be able to happen if the dome is airtight and watertight.
Factual error: The scenery is set in Berlin, following the 1989 Re-Union of Germany. Only thing is: the guards at the archive building use AK-47s, while after the Re-Union, the East German National People's Army and the West German Bundeswehr (Federal Defense Force) were joined - and the West Germans don't use any AKs.
Factual error: CDR Crighton has on his astronaut flight suit. He is wearing an American flag on his right upper arm. But the flag has the field of stars to his rear. It is a flag designed to be worn on the left arm, not the right.
Factual error: The group finds an underground sea and there is a flock of pterosaurs flying overhead. The professor identifies the pterosaurs as the missing link between pterodactyls and archaeopteryx. This is not true as archaeopteryx (and other birds) did not descend from pterosaurs but from a branch of theropod dinosaurs.