When telling Smithers he doesn't need to diaper him or burp him, Mr. Burns proceeds to belch in a similar manner to Barney Gumble.
Burns "telepathically" requesting that Smithers come back from his vacation early was similar to The Shining.
When Burns and Smithers are driving away, Burns is shown answering the phone. Later in the same episode, it shows that he never does that himself.
When Burns sits back down after making his failed phone call to Smithers, his black eye disappears.
If Burns were really 104 years old, he would have been born long after the dodo became extinct. since Burn's age is never truly confirmed, it is impossible to say if he could remember the Dodo bird.
Lisa is eating spaghetti and meatballs with the rest of the family in the first scene of the third act. This is a few episodes after she had officially turned vegetarian on Lisa the Vegetarian. Lisa's vegetarianism wasn't mentioned again until Homer's Phobia, so it may not have been considered to be canonical yet. It's also possible that her meatballs are made of some kind of vegetarian meat substitute.
When Smithers calls Homer, his phone is black, but in another scene it's blue.
At the race track, Smithers is shown to be feeding beer to Mr. Burns by the teaspoon, but in Dancin' Homer, Burns drinks an entire tub of beer without any assistance. it's possible that he became reliant on Smithers for drinking beer after the events of Dancin' Homer.
In Rosebud Mr. Burns’ teddy bear, Bobo looked old and moldy with one eye dangling after the Simpsons found it in a bag of ice, but in this episode, Bobo looked brand new and clean, but one eye was completely missing.
It is almost a certainty that Mr. Burns had Bobo repaired, though I can not explain the missing eye.