Continuity mistake: At the beginning of the show, the hospital Mr. Burns is at has a sign in front of it that reads, "Got Tumor?", but when we see the hospital again it isn't there. Additionally, the "Mayo Clinic" sign atop the building gets noticeably smaller in the second shot.
Plot hole: Dr Marvin Monroe says to Marge he was gone for a long time because he was sick. That's impossible because he died in a previous episode and you can even see his tomb in the episode where Maud Flanders dies. [It could be argued that Monroe really was sick. It was never explained how he died and no one mentioned that he really was dead. The grave, the hospital named after him were in the simpsons universe with the belief he was dead. The clip show mentioned his death but alluded any specifics. As Marge mentions she thought he was dead and so could everyone else on the show.]
Other mistake: Bart and Lisa overload the nuclear power plant to create a city-wide blackout so everyone can see the meteor shower. All the lights in Springfield blow out. Yet when we see the Simpsons sitting on their lawn watching the shower, every single window in their house is yellow, as if every room has their lights on. It is the only house shown with power.
Continuity mistake: This mistake happens twice in the episode. At first, when Homer arrives on the island, he gives the islanders some bibles and says "Knock yourself out." Ack says that they cannot read. Yet later when Homer is revealing the casino, Ack reads "The Lucky Savage-?"
Continuity mistake: When Homer writes an autograph for Ralph at the restaurant, Lisa is sitting opposite Marge, has yellow liquid in her glass and has red and brown food on her plate. In the next shot, Lisa is opposite Homer, has a blue drink and has white and green food.