I would leave them alone until the vine completely died. They will usually store well in a cool spot under your bed or in a closet. Just make sure sure you cure them first!
well growing potatoes in unfinished compost was a bust. pill bugs ate every single one. i found a few with holes in them. the bed was crawling with all kinds of bugs
so i pulled all that crap and spread 2 pounds of DE. i hope they all die.
if not ill pour 5 gallons of gas on the sob tomorrow and burn it.