I've conquered the map as both the Moors and Turks, and at the end of both games the Papal States was the only faction bar mine that remained in the diplomacy scroll.
If they have no provinces, the new Popes appear in either Rebel provinces or within the Rome region. I think if there is no Pope at present then the Papal States just continue with the units and Generals they have until a new Pope comes along.
If the Pope is killed during a turn, then if you are a Catholic faction you can notice in the Pope scroll that is says something along the lines of "that Papal seat remains vacant, a new Pope will be elected".
So I'm pretty sure the Papal States will always remain, waiting for a Cardinal to spawn himself as the new Pope in some province.
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