"Do Shut Up" could be a parody of The Young Ones, a similarly chaotic Britcom that ran on English T.V. in the early 1980s. It was known for its lowbrow characters who shared a similarly dismal house, and spent much of the show bashing each other over the head with household items.
The scene where Homer is licking bullfrogs on an island refers to the first verse of Pink Floyd's Comfortably Numb song from The Wall album. Marge calling "Hello, is there anybody in there?" over the radio even intones the melody.
Also, a similar storyline was used in Family Guy, where several characters lick frogs and go into a strange hypnotic trance, similar to what happens to Homer in this episode.
The telethon scene at the end features various FOX characters as phone operators, including Bender from Futurama,Hank Hill from King of the Hill, Thurgood from The PJs, and Mulder and Scully from The X-Files.
PBS characters who chase Homer through the streets include The Teletubbies (they fire lasers from their antennas), Oscar the Grouch (who is in his trash can), Elmo (who is sharing Oscar's trash can), Yo-Yo Ma, Mister Rogers, and Big Bird (who comes flying down, squawking angrily like a real bird). The DVD commentary reveals that in order to evade copyright, two of the characters were renamed "Oscar the Grump" and "Elmaux".
As Homer and Lisa Jr. begin to float down the lava flow, he says "Looks like this is the end." At the end of every Underdog cartoon the narrator would say "Looks like this is the end. But don't miss our next Underdog Show!"
When Homer tries to hand out Bibles when he arrives on the island, Qtoktok claims that they cannot read. However, when Homer unveils his casino, Ak is able to read the sign. It is possible, though highly unlikely, that Homer taught Ak to read before completing the casino.
The craps game is played on a roulette layout, and uses three dice instead of two. It could be Homer's own version of craps.