The usual complaint is that the original gets marginalized or removed to make room for the new. That was one of the reasons for the huge backlash against "female Thor" in the comics a few years ago - the (bad) writing mostly shat on the original Thor to get him out of the way to justify giving the hammer to someone new, who seemed to get the job solely because she was female. And as I previously mentioned, there was backlash when DC introduced a new Hispanic Blue Beetle mostly because they shot the old (fairly popular) Blue Beetle in the head and killed him more or less solely to get him out of the way so the name would be free.
It's not even necessarily limited to race/gender, though - comic fans were PISSED back in the 1980s when Steve Rogers stepped down from being Captain America and John Walker took his place (though fans being pissed was part of the point of that storyline, because they always intended for Rogers to take the name/costume back). The same sort of happened with Jean-Paul Valley replacing Bruce Wayne as Batman in the 90s Knightfall storyline (and again, fans being pissed was sort of the point - you were supposed to be annoyed that this guy was taking Bruce's place, so you'd welcome Bruce back when he returned). And major changes in general can annoy people even if the character themselves don't change at all - the storyline in the 90s where Superman turned into a living laser sort of energy being who could shoot lightning (and later split into two separate people) annoyed a lot of fans, even though it was still Clark Kent.
Comic fans are generally much more open to the idea of "diversity" legacy characters when the new character doesn't hurt the old one. A large part of why fans like Miles Morales and Spider-Gwen is because Marvel didn't kill off Peter Parker to make room for either of them, and they generally co-exist (and interact with each other). A lot of people liked the Young Avengers version of Hawkeye because she actually hung out with and trained with the original (something Marvel has implied they're probably going to do in the MCU as well).
Well removing a character is stupid. They can just create a new character while retaining the originals. ---