Continuity mistake: Whilst Fry, Leela and the professor are on the log near the waterfall they are very close to the right side of the river. A few shots after Gunter puts the hat back on his head you can see they and the log are now all the way over to the left side of the river.
Continuity mistake: In the scene where Fry and the professor is trying to make Gunter choose between the banana and the hat, the distance between Fry, the professor, and Leela changes noticeably between shots. First Fry and the professor are wide apart with Leela behind Fry. Then after Leela hears the robot-house's boat approaching, they're all suddenly down by the lake standing next to each other.
Continuity mistake: When Gunter is trying to figure out where the hat goes, the banana disappears and reappears between shots. When Gunter finally figures out the hat goes on the head the banana is nowhere to be seen, yet when he needs it to do some calculations the banana is on the ground right next to him.