Victor Frankenstein states that hair and fingernails continue to grow after death. While commonly believed, this is false. As the skin becomes dehydrated, it recedes, exposing hair and nail tissue that was already there.
Just before the big announcement of Victor starting medical school, Victor is dancing with Elizabeth. They stop, and the camera focuses for a few seconds on Baron Frankenstein. It then focuses on the crowd, and Justine is now standing next to Victor and Elizabeth is not near him.
During the movie, the scars on the monster's chin and lip change. First his lip is split, later it's healed, but now he has other wounds on his chin. Than his lip and chin are healed, with no scars. But later you see big scars on both his chin and lip.
After Elizabeth lights herself on fire and runs down the hallway, she jumps off the staircase to the landing below. As she is falling towards the floor, you can see the floor is darkened with "soot" in the exact spot she is about to land before she hits. This shows evidence of previous takes but the floor was not cleaned between takes revealing the already scorched floor.
The goof items below may give away important plot points.
After the murder of Elizabeth, Victor brings her body back to the family mansion outside the rainstorm. When Victor enters carrying Elizabeth and is stopped by Henry, his hair is soaking wet from the rain. In the next shot, he enters the laboratory and his hair is completely dry.