Visible crew/equipment: When Mulder and Scully are talking to the owner of the trailer park and he is berating Mulder, you can see the edge of the lens in the upper left corner of one shot. This is because the lens used was of a wide angle almost fish-eye type and the edge of the lens itself can occasionally creep into the frame on most cameras.
Revealing mistake: The answering machine at the end appears to not be plugged into any phone line. The shot shows the phone jacks on the right side of the machine are empty. In order for an answering machine to work at least one of those ports would need to be hooked to the phone.
Plot hole: It is virtually impossible that such a conspiracy-experienced, cautious, and wise person as the Second Elder would be so naive and careless to just let the "Dr. Openshaw" faceless alien to his house, knowing from the CSM that Dr. Openshaw had died that evening.
Factual error: At the beginning of the "Musings of a Cigarette-Smoking Man" episode Frohike says that the CSM appears when Trotsky was assassinated, i.e. 20 August 1940, but he does not claim that this is an exact date of birth. However, he further says that during the Nazi-Soviet Pact (1939-41) the CSM's father "kept NKVD informed on American plans to enter World War II" and for this reason he was executed before the CSM learned to walk. This means that his father could not have been executed earlier than in 1941. Assuming that babies learn to walk between a few months and 2 years, the CSM could not have been born earlier than in 1939. It means that in 1953 he should have been a young teenager, and not an adult man, as he is presented.
Visible crew/equipment: Mulder talks to the man from NICAP on the other side of the fence. When the shot is looking at the man, film lights are visible reflected in his glasses. Since he is looking at Mulder, there should be lights on Mulder's side, but there are none.