Typically, peanut butter is considered shelf-stable and can be stored at room temperature without growing mold-forming bacteria, but this doesn't mean it will stay fresh.
This is especially of concern with natural peanut butter, which is made with no stabilizers to preserve the texture, and the nuts' oils can quickly separate unless chilled.
Storing natural peanut butter in the refrigerator will prevent it from separating and keep the oil from going rancid, thereby preserving the freshness of the contents in the jar.
In hot and humid climates, oils can turn rancid quicker, and a refrigerator may help peanut butter stay fresher. A short shelf life should be safe if you quickly go through a jar.
Storing peanut butter in a refrigerator will significantly impact its texture. Set it at room temperature or give it a few seconds in the microwave to make it smooth again.