The boardgame Bart is playing in bed is Stratego. He uses the pieces as action figures and says "I swear I don't know where the bombs are. Only the miners know that.". Which goes with the rules of Stratego, only the miners can defuse bombs.
The Itchy and Scratchy episode featured in this episode "Planet of the Aches" actually refers to the second film in the series "Beneath the Planet of the Apes", which features a race of mutant humans with enlarged brains and telekinetic powers. The future Itchies killing Scratchy by using telekinesis to cut him up with a lot of sharp objects is a parody of the ending to Carrie.
In the far away shot of the Springfield Pool Mobile, a kid overlaps with the side of the vehicle, making him look like he lacks legs.
The portable TV that's in Bart's room disappears in some scenes. Being a portable TV it's not out of the question another family member borrowed or removed it.
When Bart puts away the script for his play you can hear the sound of the closing drawer, but it remains open.
As Bart adjusts the aim of his telescope, a white-colored section of the telescope turns black for a frame.
Nelson states epidermis is another name for hair. It actually means the outer cells of the human skin. This is most likely a result of Nelson's ignorance than an actual mistake. Of course, it's still true that it's showing.
In the HD remake, Matt Groening's name is misspelled as "Matt Gorening"