Like anything else, belts should be stored in such a way that they are neat and accessible.
If you like keeping your belts in your dresser, you may just need to buy some drawer dividers or shallow, smaller boxes, to compartmentalize the space. Then you can loosely coil your belts and put them into the containers in a single layer; they won't have room to spring to life. The boxes can also be kept on a closet shelf.
The other alternative is to hang the belts from hooks -- the back of the closet door or a closet wall is the perfect place for this. If a garment comes with its own belt, it is best to remove it before you hang up the garment, otherwise the belt could pull the fabric out of shape.