Critical Research Failure: When Ned attempts to pass off Green Eggs and Ham as the book he'll use for his book report, he states that the person who doesn't like Green Eggs and Ham is Sam when in actuality, Sam was the one who liked them and was trying to get the unnamed character to eat them.
Cookie as well, especially around Season 3, where he began to take interest in Lisa Zemo after she had stopped having her crush on him and changed her appearance. His tendencies to overreact to a lot of things as well as screwing up a lot (especially causing Ned to fail his exam in Study Hall) didn't help the cause either, reaching Took a Level in Dumbass red flags.
Gordy also qualifies, as he was elevated from a recurring role in the first season to a starring role in the second and third seasons.
Family-Unfriendly Aesop: The Class Clown episode. While it is true that being amusing can lighten the tension in a School Environment, the episode encourages the viewer to clown around, even if it results in you getting in trouble.
"Funny Aneurysm" Moment: The Halloween episode where Principal Pal is presumed dead after going into the Haunted House. Pal's actor, John Bliss, died a year after the show ended.
The show frequently uses puppets and unconvincing People in Rubber Suits to portray various animals, and sometimes rather obvious dummies representing characters doing painful stunts. It seems that this is largely done on purpose, considering the cartoony nature of the show to begin with.
In the Pep Rallies episode, Cookie jumps onto a powerful trampoline and is launched through the gym's roof and lands outside the school. This was done using a computer model. However, immediately after Cookie crash landed outside of school the model flipped into an upright position.
This is actually invoked in one episode where Moze thought she'd knocked out two of Coconut Head's teeth. The "teeth" are obviously pieces of white breath mints that have been cut to resemble teeth. Since audiences were used to the terrible effects, the audience just assumed it was his teeth. Then it was revealed that it really was breath mints in-universe as well.
At the end of the "Guide to: Notebooks", Sweeney is hanging from a harness in his classroom. Gordy and Ned are in the room with him. Gordy swipes a caramel apple out of Sweeney's hand, and proceeds to spin him around like a tornado. As the screen goes to an Iris Out transition to go to commercial, you can look through the editing effect of Sweeney rapidly spinning around, and see Don Creech, the actor who played Sweeney, completing his first rotation, flailing his arms and legs, and screaming in a panic.