It will spoil/go rancid. Even the 'non-natural' big brands like Jif/Skippy taste and smell 'off' if they sit out for a few weeks after being opened.
There's no winning here, because putting it in the fridge makes it hard to spread. The best approach is to take out a week's worth or so and put it in a small/airtight container. Leave that out but put the main jar in the fridge.