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.)
Waylon Smithers was designed to be "Mister Burns' white sycophant," as it "would be a bad idea to have a black subservient character" putting up with the abuse that Burns doles out. (Quotes are from David Silverman, one of the show's main producers.) Ironically, Smithers is colored as an African-American in his first appearance The Simpsons: Homer's Odyssey, due to color stylist Gyorgyi Peluce's misinterpretation of the new character. The mistake was quickly caught and corrected starting with the immediate next episode The Simpsons: There's No Disgrace Like Home.
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.
When Homer bowls a Perfect Game, the Bowlerama Employees both take out keys to unlock the button that releases the Balloon. But the keyholes are too close together. Double Key systems are supposed to ensure that it is impossible for one person to carry out the action, so the keyholes should be much further apart.
Fat Tony's real name changes from episode to episode, but this is probably intentional because it is common for Mafia hit men (in popular culture, and possibly in reality as well) to have many different identities.
"In one episode they say this, but then in another episode they say that, and in yet another episode they say the other." As this is an animated comedy series, the emphasis is clearly on laughs rather than complete verisimilitude. Efforts are certainly made to create a vaguely consistent setting in which mostly consistent characters live and work, and many episodes refer to each other, but rigid consistency of every single detail in all episodes is unnecessary. In many episodes, the fact that something is inconsistent is the express point of a gag. Our general rule is that each episode is expected to be consistent within itself, but intra-episode inconsistencies are not being listed. There can be exceptions for unusually noteworthy matters, e.g., inconsistencies repeated in multiple episodes (such as the hair and skin colors of secondary characters, and the layouts of the main landmarks), drastic changes to a character's nature (such as Ralphie's school status, Jasper's abilities, or Milhouse's hair color), or something with an interesting anecdote behind it (such as Smithers' skin color).