Let's cover the first thing, which is to clarify what is fan fiction. It is stories created by fans, derived from the work of other writers (referred to as, "derivative works"). The problem comes about when the work of another writer is still under copyright protection.
Next thing is, just because the writer of a derivative work does not make a profit from something doesn't mean they aren't violating copyright protections. Express consent from the copyright owner is required and a person may be sued for actual or potential market loss, as well as any direct or indirect profits obtained from the presentation of the derivative work. This comes as a bite to any of those websites that opt to host such works, as they can be sued for all gross income (i.e., before expenses) obtained from branding, advertising, or other profit activities gained by hosting the reader-generating derivative works.
So, really, if you don't obtain express consent from the copyright owner, why bother with the potential hardship? There are plenty of authors who have no issue with not-for-profit fanfiction, but there are also some (like John Grisham) who will sue your pants off at the first opportunity.
