Somewhat intuitively, a creature that consumes fungi is a fungivore ( from Wikipedia ).

Most creatures that eat fungi are, like us, omnivores in that they don't exclusively consume fungi. The wiki article for this term probably gives you a better idea of the variety of species that do this. I didn't realize this, but there are also some (mostly amoeba) that are bacterivores bacterivore . So I would guess that this naming convention of "kingdom+vore" holds pretty broadly.