It's not required to keep peanut butter and jelly refrigerated as the peanut butter and jelly will last up to 30 days at room temperature after opening the jar of peanut butter and jelly.
However if you do refrigerate the peanut butter and jelly it will last up to 6 months.
So although you're not required to refrigerate peanut butter and jelly it's best to refrigerate the peanut butter and jelly to make it last longer unless you plan to eat the peanut butter and jelly within 30 days.
Usually the jar of peanut butter and jelly only lasts around a week at most around here before it's always used up so I never refrigerate it and keep it in the cupboard.