This can easily switch to hatred of the previously loved celebrity, and result in attempts at violent attacks, one notable incident being the death of Rebecca Schaeffer by a stalking fan in 1989.
This is somewhat related to the concept of parasocial interaction where audiences develop one-sided relationships with media characters and celebrities.
Otaku is a Japanese term for people with obsessive interests.
In Japan, the term is normally derogatory, a connotation lacking in English, where it generally refers to people in the anime and manga fandom.
Supporter is a synonym to "fan" that predates the latter term and is still commonly used in British English, especially to denote fans of sports teams.