Springfield Elementary School is yellow in some episodes and white in others, sometimes changing color multiple times in the same episode (later episodes corrected this continuity problem by making the school's official color yellow, and including a picture of the "old, white" school in Principal Skinner's office.)
When wearing her normal green dress, Marge wears a red necklace. However, often she is shown for a single frame or two wearing a white one (generally this happens in close-up shots of her head.) This happens in numerous episodes.
In "The Itchy And Scratchy And Poochie Show" episode, in the comic store at the autograph signing, the Rainer Wolfcastle poster in the background spells his character's name as "McBane." In all other episodes it's spelled "McBain."
In "Two Cars In Every Garage And Three Eyes On Every Fish", Mr. Burns tries to bribe the nuclear inspector by stuffing wads of cash in his pocket. When the inspector leaves the wads of bills in his pockets are gone.
... almost everything. This is a comedy animation series, and while efforts are certainly made to create a vaguely consistent setting in which mostly consistent characters live and work, the emphasis is clearly on laughs. Individual episodes must be consistent within themselves, but rigid continuity between episodes is unnecessary. Indeed, additional humor has been wrung, during the life of the show, from the very fact that the episodes are inconsistent with each other. No further "In one episode they say... but then in another episode they say..." goofs can be accepted.
When Waylon Smithers makes his first appearance in "Homer's Odyssey", he has orange skin. He has the standard yellow flesh in all subsequent appearances. This was actually a change that the creators made knowingly. Smithers was mistakenly animated with the wrong color and was made an African-American by Gyorgi Peluci, the color stylist. David Silverman has claimed that Smithers was always intended to be "Mr Burns' white sycophant," and the staff thought it "would be a bad idea to have a black sub-servient character" and so switched him to his intended color for his next episode, the fourth episode aired.
Jasper confesses to being blind when viewing the premier of the Poochie episode, but could see perfectly well in a number of episodes (i.e. watching the Tatum fight at the Simpson house and walking down the street when Smithers shot him in his wooden leg.)