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.
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.