"I don't want it anymore" should be accepted.
EDIT: It shouldn't be accepted check Nitram15's comment :)
No, it shouldn't be accepted. "i don't want it anymore" would be: "Non lo/la voglio più" The pronoun "ne" has a meaning "of something", just think of this conversation:
Do you want some of these apples?
No, thanks, I don't want any of it.
Vuoi di queste mele?
No, grazie, non ne voglio.
What you want to say would look different. Look at this conversation:
Do you still want this apple?
No, thanks, I don't want it anymore.
Vuoi ancora questa mela?
No, grazie, non la voglio più.
So, remember: "ne" can only replace structures with "di+anything". And it can stand in idiomatic phrases like "andarsene", for example, which means "to leave", and is conjugated like this:
(io) me ne vado
(tu) te ne vai
(lui/lei) se ne va
(noi) ce ne andiamo
(voi) ve ne andate
(loro) se ne vanno
I'm quite sure this could also be "I don't want (any) more of them." Should I report it?