At the beginning when Mr. Farnum hides the recorder on the lower shelf, the bust is on the left side of the brown books. When Mr. Farnum comes back to switch out the tapes, the bust is on the right side of the brown books.
The first time Jeannie turns Mr. Farnum's film to pink goo, he opens the tool chest and it is already goo before she blinks to change it.
The marriage certificate says that Tony was married in 1947 but this would be impossible because he would be in his mid 40s when he goes to the moon. Nasa would not have astronauts that are that old.