Revealing mistake: When the professor is trying to find the gremlin, he is in the room with the movie playing. The projector that is playing the movie is actually moving backwards. (00:44:35 - 00:47:35)
Revealing mistake: The robber who kills Billy's mother is shown cutting her throat but with the flat side of the blade. Moreover real blood doesn't ooze out so thickly, indicating it was makeup.