Continuity mistake: When the robot at the start is dying, the doctor opens a panel to see his "heart beat." After a couple of other shots we see the panel again and this time it's closed without anyone touching it. It's also important not to confuse the doctor closing the robots eyes with this panel.
Revealing mistake: In the final battle, Bender has fry trapped into the seat inside of him, and he drives over a ramp and nearly smashes into the original were-car mid-air. But when the were-car lands on the same ramp that Bender took off from, we see that the furnace is literally touching the end of the ramp. How did Bender get up enough speed to launch himself into the air, especially accidentally (he would have had to turn 90 degrees to get onto the ramp) with the furnace that close? Also at the start of this scene, we see the were-car crying in front of the furnace, and the ramp is nowhere to be seen.
Continuity mistake: When they find Bender on the ground after being hit by the were-car Fry asks if any of them have any liquor you see all of them reaching out a bottle of liquor. When Bender grabs one of the bottles it cuts and you see all of them standing back as if none of them were handing him anything.