When Fry sells his clothes, he returns to the Planet Express Building naked. In the next scene, he is shown putting his clothes back on, despite having sold them. However, he does not put on his jacket, which he has on after the trip.
When Farnsworth is inquiring about the National Ray-Gun Association, Fry's girlfriend Michelle can be seen walking in the background. This, however, is impossible, as Michelle was unfrozen in the episode The Cryonic Woman, which is set a few months later.
When Bender has the dream with zeroes and ones, he stated he saw a two. He should have also seen the four in the dream.
It's stated that Nixon cannot be elected due to the rule that no body can be elected twice, this isn't true. There is no actual clause in the US Constitution that is worded like this. Also, if Earth is a nation of its own, why would all of it be using the US Constitution?
In the episode "A Farewell to Arms", it's revealed that Fry has been wearing the same pants for a thousand years, which is impossible if he sold his clothes to the pawnshop. It is possible he bought them back at a later time however.