If you are buying natural peanut butter, it will go rancid after a couple of months at room temperature, or even less time than that. And the oil will float to the top so you have to stir it in all the time. Refrigerating solves both those problems. If you always finish a jar of peanut butter within a couple of weeks, neither of those things will be an issue. I buy natural peanut butter and sometimes don't finish a jar for six months, so I always refrigerate it after opening.
If you are buying one of the homogenized with sugar added brands, you don't need to refrigerate it. It won't go rancid and it won't separate at room temperature.