As I've been considering the topic, I've been contemplating what it is that I personally dislike about fanfiction. Among the field of problems associated with it (most of it is badly written and/or wish-fulfillment, much of it violates creative intent and/or copyright law, etc.) is one that hasn't been mentioned yet:
It implies that the character is more important than the author.
I've read almost every book that Diana Wynne Jones has ever written. She seldom writes about the same characters twice. I'm very fond of some of those characters. I'd enjoy reading more about Chrestomanci (who does appear in several short stories and a few books), or Rupert Venables of Deep Secret, or finding out what happens next to Thomas Lynn and Polly of Fire and Hemlock.
But I don't go searching for fanfic about those characters (there's probably some out there) or write to Ms. Jones begging her to do another book about Rupert.
And part of that is because I trust Ms. Jones as an author. I trust that what she writes next will be what she's inspired to write, and that it, too, will be good just as her other books have been good. It doesn't have to be about the same characters.
There's something ... suffocating about the human demand for more about this character. Or this universe. Or this team. It leads to a lot of bad writing: writers flogged to make what will sell, rather than what they are driven to make.
This isn't just about fanfiction; it's about professionals writing their own creations, too. Glen Cook has written I-don't-know-how-many Black Company books now. If all his books were equally popular, he'd've stopped writing Black Company by book five (if not sooner). But Black Company is what sells, so he flogs his muse to produce Yet Another Black Company novel, and it makes him ten times what any novel in a new series would. So that's what he does.
But there are novels in his head -- new stories, in new settings, about new characters -- that will die with him, because not enough people are willing to take a chance on them. Because they won't say "Hey, I loved this series by that Cook guy -- maybe this other novel by him will be good, too."
When I was in fifth grade, I read a dozen or more books about Oz. The last one I read was the sixth novel by Frank L. Baum. It ended with the statement that there would be no more visitors to Oz, no more books about Oz. I was shocked. How could this be? I'd read way more than six Oz books. And I finally thought to check the spine: most of those books had been written by other authors. It had simply never occured to me that might be the case.
What is it about us, that we value the creation more than the creator? Why do so many people say, "That was great! Can you do another one just like it?"
Why don't we want to see what else they can do, that might be new, and fresh, and different and maybe -- just maybe -- even better?