Deliberate mistake: When Bart is in the genius school, the teacher asks him if he knows what happens if he adds an acid to a base. He replies yes, adds them anyway and the lab explodes and a green goo comes out, turning Bart green. If you add an acid to a base, you just get salt and water.
Continuity mistake: The guardrail at the Springfield Gorge keeps changing between shots when Bart is about to make his big jump; in the first wide shot there is a short rail opposite the "stone ramp." Then in the Bart-POV shot from atop the ramp, the rail is much longer, and the parking lot and picnic area behind it have disappeared. And when Homer is actually jumping the gorge, the rail disappears altogether.
Continuity mistake: In the end of the episode when the family has reclaimed the house and are cleaning up the living room. As Homer mentions about letting the Cooders stay, Bart tells Marge to get him away from the window. She is standing right next to the couch in the lounge but the very next shot she lowers her hands and pats the couch yet is now in the family room. (Note: There was a few shots that showed there was two couches so the mistake is with Marge suddenly moving not the couch.).