Pretty sure they won't keep growing without the foilage. If the ground is dry enough, they should keep for a while - best place IMO to keep them. If you get a heavy rain, pull them out and dry them with the dirt on, then store in dry, dark place. They keep better & longer with dirt on.
We lived next to a potato farm few years back...that was a bonus! Wait till its twilight, then scoot between the barbed wire and, umm, harvest
Had to dive down between the plants sometimes when the farmhands made a late trip around in their ute...lol. Any of our visitors would bring their sacks along too so we'd all go for a late night forage.
Always left the dirt on them, stored them on newpaper in a cardboard box in a dark cupboard. Never went lackiing spuds that year - for about 4 months we could get them, and the rest stored a good 2 or 3 months. (Cold climate here). Just had to chip all the caked dirt off then soak and scrub to cook 'em. The babies were delish, and later in the season the big monsters were really great for a mash - one spud fed 4. (They grew them commercially for McD's).
You should look up a previous post by KYH about growing them vertically - interesting stuff.
pulling potatoes 
