Refrigeration, no -- no need. But beyond that, storage is largely a question of what you're trying to do.
If you want perfect steamed rice, you need your rice to be stored airtight, in a not-too-hot place, for not very long. Rice must be fresh, or it dries out and loses its sweetness. For perfect steamed rice, buy new crop rice that is clearly dated and recent, keep it tightly sealed and out of the light, and use it up within 2 months at the outside. That kind of rice will cost about 10x more, give or take, and is worth it if you like steamed white rice.
If you're using rice in other ways, e.g. flavored with something, baked with stock and butter, and so on, it doesn't make nearly as much difference -- almost none, in fact. Just make sure you store it tightly sealed and out of bright light or heat.
Note that cooked rice will get hard and nasty if refrigerated. To store cooked white rice, seal it tight in a tupperware sort of thing and freeze it solid. To reheat, nuke it gently, then fluff with a fork.