Fanfiction by definition is not owned by the writer. It is derivative work using copyrighted characters, places, and histories.
"Copyright law is designed to encourage authors to be creative by rewarding their efforts and protecting their work from others who might profit unfairly. This right must be balanced by society's need to have others not be limited by previously published protected works. There is not a clear 'right' and 'wrong' side in the battle between copyright owners and fan fiction writers." (Chilling Effects Clearinghouse)
EAB & Fan Fiction
Considering the fact that I am a fanfic writer, I obviously won't discourage it in others. It was a wonderful, soothing outlet for me during high school and helped me keep my writing skills sharp without the creative pressures that I currently enjoy. I think that fanfiction, properly written, can be one of the most glorious things in the world. But I also know that it can be some of the most mind-numbing, senseless dribble, and that reading fanfiction to the exclusion of published books can stunt a growing writer. Published books are held to a higher standard (spellchecking, constant editing, spellchecking, spellchecking, spellchecking ...), and as one who has written her share of dribble and read more than enough, I urge you to befriend your local library and take fanfiction like you would McDonalds -- for a treat, but not as a steady diet.
I will not read any fanfiction based on my work. For one thing, I've heard that some authors have been sued by fanfiction writers for stealing their ideas (crazy and stupid, but people are crazy and stupid). And even though I know my readers are too nice to do anything like that, I have more aesthetic reasons too: first, I've never enjoyed reading fanfiction that's based on a written work (like Harry Potter) because I'm very sensitive to writing style; second, I'd be far too conscious of factual mistakes to enjoy myself; and third, I really don't want to see Strife and Deathlock making out.
If I were to consider reading fanfiction in any way, it would be in contest format, and I would have a lawyer-approved Terms of Entry to absolve me from any sticky issues down the road.
World of Warcraft Fan Fiction
Dustfire. [June 9, 2008] Short story based on my raiding main, written for my brother's in-guild writing contest. "Use the short-story format to introduce the reader to your character and give essential background information."
Anime Fan Fiction
I have not decided yet if the benefits outweigh the risks of posting my anime fanfiction here. As a professional writer, it could be argued in court that posting fanfiction benefits me in material ways.