Continuity mistake: When Quatermain arrives in London, he is greeted by Reed who leads him down some stairs. In the shot where they're going down some spiral stairs, for some reason you see a different actor (who is clean shaven) posing as Reed, going down the stairs holding the umbrella in his right hand. Then in the next shot you see the "original" actor who has a slight goatee and is holding the umbrella in his left hand.
Continuity mistake: The invisible man is seen in one shot with cream covering the majority of the front of his face but it ends towards the middle of his head. When he is in the automobile, the cream is now completely covering his head and neck. Then, when they go to see Dorian Gray, only his face is covered with make-up again.
Continuity mistake: When the Phantom's men ambush the League at Dorian's residence Sawyer is seen joining them in revealing themselves and aiming at the League (you can tell it's Sawyer by his clothing and weapon). Then during the Phantoms speech Sawyer reveals himself again as if he was never there.
Factual error: In the first few scenes they show some newspapers. They show Tuesday April 10, 1899, then Thursday May 18th,1899. If April 10th is on a Tuesday, then May 18th can't be on a Thursday, it would be on a Friday. April 10, 1899 was actually a Monday, but May 18th that year was indeed a Thursday.
Plot hole: When the team plays back the record, it contains an extremely high-pitched tone which triggers the crystal sensors on the ship. But the speech on the record is low quality. This low quality would be the result of extreme frequencies being lost; but if they were, the trigger tone (a VERY extreme frequency) would be lost too. If the League's gramophones are capable of recording and playing such a tone, they should also be able to record and play back the speech without any quality loss.