The lunar rover has a radio with speaker holes. In the vacuum of the moon no sound would be transmitted.
Moon Patrol officers threw Bender out of door to moon's surface. All of them should have been blown out the door by explosive decompression.
The United States flag should've been completely white due to the sun, especially after over a 1000 years.
Leela and Fry find the site of an original moon landing but the lunar module is still intact. In reality the body of the module launched back into space leaving only the legs behind. However, it clearly states inside the lander "Lander returned to this site by the Historical Sticklers Society".
In the Planet Express commercial during the opening scene, the delivery boy is wearing a banged-up spacesuit with a helmet while delivering a package on the surface of the moon, but the recipient of the package is not wearing a helmet, or any protective gear. Without a spacesuit, he would have died instantly.