It's probably best to store your rum in the refrigerator, honestly. I've never really heard of leaving the rum out, as it's best served and kept chilled when you're not drinking it.
Well, it would be best if you put it in the fridge. You can also keep it in room temperature but I think it works best in the fridge.
There's not any reason to, unless you just want to keep it cold. Storing it with the rest of your liquor is fine.