At the begining of the episode, the Statue of Liberty is facing west towards New Jersey when it should facing east towards Long Island and the Atlantic Ocean.
Bender is usually portrayed as starting to rust and malfunction, if he goes too many hours, without alcohol. Yet, at the end of this episode, he's completely fine, when Fry digs him up (despite the fact that Bender had spent over a thousand years, deprived of alcohol, when he was in the ground).
The episode explicitly stated that time travel has landed the crew in Roswell, New Mexico in 1947. However, there were no atomic tests performed by the USA in 1947, and never any tests in New Mexico in any year.