Clearly, the story makes you unable to ignore how the beloved toy is slowly coming apart, more or less voluntarily sacrificing itself for the love of this boy. This is, by itself, fairly hard to take. But as this quote makes so clear, what's even harder to think about is how this rabbit, in its devotion, is actually a proxy for a loving parent. The story asks you to think about the boy from that perspective: through the eyes of his parent.
To read a story from the perspective of a dying protagonist is to entertain the idea of yourself dying -- always a wrench. To read a story in which a parent dies is to think about your own parents dying. The Velveteen Rabbit story combines both of these into one perspective.
S: [sarah weighs in: let's do this conversationally!] so then, the story describes the way in which parenthood/realness are both "about" death; about the inevitability of mortality. Having children puts you in the stage of life that is closer to the reality of death.
Of course, in the novel (if I remember, it's been a long time) the rabbit is explicitly *not* the parents, who DO NOT stay with the sick child, and who are the ones who burn the bunny.]
B: Right, the rabbit is not literally the parent -- the boy has parents. But, as so often in these kinds of stories (cf. Harry Potter, Rudyard Kipling, the Dark is Rising, Weetzie Bat, and just about everything else), the literal parents are absent, unfeeling, or otherwise not worth worrying about for purposes of what's important to the story. Basically, they serve as formal placeholders, while the engine of feeling that actually drives the plot (even if those feelings are, essentially, parental feelings) is located somewhere else. That just seems to be something about how child-stories function, don't you think? They are allergic to literal parent/child relationships.**
P.S. The Rabbit doesn't get burned. His fate is stranger than that, and just as tragic but in a different way:
While awaiting the bonfire in which the Velveteen Rabbit will be burned, the Rabbit cries a real tear. This tear brings forth the Nursery Magic Fairy. The Rabbit thinks he was real before, but the fairy tells him he was only real to the boy. She flies him to the woods, where he realizes that he is a real rabbit at last and runs to join the other rabbits in the wild.
The subsequent spring, the boy sees the Rabbit hopping in the wild and thinks he looks like his old Velveteen Rabbit, but he never knows that it actually was.(wikipedia)
The rabbit is not killed, but transmuted into a different plane of existence, a different kind of story. He is still "alive", but alive in the way "real" rabbits are: they don't talk to us, let alone love us. They are not our toys. Having to realize this -- having the story lead us on a path back to the actual world in which animals are alien to us -- is one more reason never to tell this story to children!. ;-)
** One exception I can think of: Danny the Champion of the World, by Roald Dahl