This episode marks one of the few "Breaking the Fourth Wall" moments in the series; when Carl unintentionally offends Lenny, Barney, and Homer with a question about their personality flaws for Moe to relate to regarding his ugliness, all four of them break down crying. Carl then turns to the audience and states, "See, this is why I don't talk much."
The soap opera that Moe got cast in was called It Never Ends, whichis a reference to the longevity of soap operas.
The plot point with the actor who plays Dr. Tad Winslow demanding a raise only to be fired may be a reference to the then-recent pay dispute behind the scenes of The Simpsons, when Maggie Roswell demanded a raise that the FOX network was not willing to meet, leading her to temporarily quit the show.
The word, "fudge" was censored for some weird reason. Possibly for comedic effect.
When the Simpson family get on the car, they wear pajamas, but when they arrive to Duff Days, there wear their daily clothes. Also: the pets are missing.
After Homer puts in the cassette tape a fifth toe briefly appears on Lisa's right foot.
Moe acts surprised to learn how ugly he is, but in "Dumbbell Indemnity" he was already aware of his ugliness.
If he couldn't breathe with the sticker on his face, how was Duffman able to talk? He might have taken a big breath beforehand.
It's odd that Moe and Homer read so far into the future of the show's plotlines without noticing that Moe's character wasn't actually absent. They probably stopped as soon as they saw that part.
Bart already knows what it's like to get drunk, as seen in "Homer vs. The Eighteenth Amendment" when he got beer sprayed through his Irish horn and he swallowed it. As he was actually drunk, he may not remember it very well.
One of competitors kicks out his drunkard Larry in the "Beer–Tender of the Year" contest. Actually, Larry is Moe's drunkard (unless Larry likes to frequent bars other than Moe's Tavern).
Only Marge, Skinner and Apu are in the "Duff Designated Drivers's Rockin' Fun Zone" in the first scene. In the next scene, the zone is much bigger and there is also Kirk inside of it.
Homer took out the car door handles so the kids couldn't open them from inside the car, but at the festival, they open the door normally. He probably gave them back after parking or when they were almost there.