There's no need to refrigerate peanut butter and I've never refrigerated peanut butter ever and just leave it out on the counter or in the cabinet and it's still good to eat even when opening it.
Even opened jars of peanut butter are still safe for up to 3 months even when left out of the refrigerator as long as you keep the lid on the jar after opening it and while storing it.
I've actually kept an opened jar of peanut butter for 6 months when I forgot about it in the back of my cupboard behind some other food.
The peanut butter was not as fresh but it was still good even after 6 months of being opened but still sealed with the lid.