In the coal mine scene (when David picks up his father) while they're waiting for the train to go by, you can see that several cars are marked for railroads (Penn Central, Conrail) that didn't exist in the late 1950s.
Mr. Cleary uses the term "quantum leap" in its modern colloquial sense, meaning a large increase. Dictionaries put the first use of the term in this rather than its original scientific sense around the late 1950s or early 1960s, so it's highly unlikely a French teacher in the 1950s would use it that way.
When Van Kelt grabs McGivern from behind and administers a noogie at their introductory scene in David Green's room, McGivern elbows Van Kelt and escapes his grasp, leaving his overcoat in Van Kelt's hands. He then darts out the room. In the next cut, a long shot from down the crowded hall, McGivern is waving his overcoat over his head as he runs away from the room and is chased by Van Kelt.
David is right-handed, so from the receiver's point of view the ball should rotate counter-clockwise when he throws it. At the end of the first game against Winchester, last two shots of David's game-winning touchdown pass, with the ball approaching the camera (i.e., the receiver's point of view), show it rotating clockwise.
When David is having dinner with the Dillon's Tom Keeting is introduced to Mr.Wheeler as he stands up he is wearing glasses. As he is about to sit down his glasses disappear. When it shows him sitting in chair his glasses are now on.