A classic example would be Mickey Rooney's yellowface role as Mr. Yunioshi in "Breakfast at Tiffany's" (1961).
https://en.wikipedia.org/wiki/I._Y._Yunioshi
It might be technically possible to excise the Yunioshi scenes from the film and still have something that holds together narratively. It's been a while since I've seen the movie and I can't remember if he's got any crucial plot role.