As Carol and Max were walking through the desert, Max sees a giant sheepdog in the distance. When asked about it, Carol responds, "Oh, It's that dog. Don't feed it, he'll just follow you around." The dog is never seen or mentioned again.
Also, that one scene where Douglas is petting a cat.
Potentially lessened by the fact that none of the characters (including Douglas himself) react in a way that implies that Douglas is in pain. If anything he reacts more with annoyance. However, this could go the other way, and the disproportionate reaction might make it seem even *more* unsettling.
Also Carol trying to eat Max in a fit of rage.
Alexander warning Max not to tell Carol that he isn't a king.
Granted, Spike Jonze himself stated in an interview that it wasn't meant to be a kid's film, but a "film about childhood".
Then again, Maurice Sendak has gone on record to state that the film is absolutely fine for kids "and anyone who doesn't like can go right to hell." So... there's that. Although, to be fair, the original was meant as a children's story, and Sendak has gotten more than his fair share of this trope, to the point that some teachers wanted to ban them.
The Woobie: Alexander. Arguably everyone except Claire, her friends, and Max's mom's boyfriend, who aren't developed enough to qualify, and the teacher, who is kind of scary.