My first thought would be to actually define existence!
In mathematics, you always have to make sure to communicate the "Universe of Discourse," i.e., the set of all objects in which we are checking for the existence of a particular object. For example, before checking for the existence of "Harry Potter," I COULD define the universe of discourse as "Characters in a book by J.K R", of which Harry would definitely be a subset.
If, however, I set the U.o.D. to be "People", then I could make the argument that Harry Potter is in fact a "person," though this would again depend on whether my definition of person depends on a physical existence. Since "person" is an English word, as well as an "idea" about some kind of personhood, I would have to conclude persons are not limited to the material universe of discourse.
If we do NOT include this foundation, then we run into the same problems Bertrand Russell saw in Cantor's (Naive) Set Theory. If I define the set of all fictional objects such that they do not exist when they are fictional, I believe we set up Russell's Paradox. Currently, mathemeticians do not assume that "for every property, there is a set of all things that satisfy that property," but rather that if you have a set, then a any subset of it defineable with "first order logic" EXISTS.
More info/quotes from here: http://en.wikipedia.org/wiki/Russell's_paradox
Based on that one might even be able to say that Fiction would have to exist as a subset of all things that I believe to exist. Whether that set of itself exists is another question entirely.