I've done some web searches on this and I've seen a lot of different answers, some say no permits needed and it is a free for all out in the wilderness. My question however is for someone WITH a permit, if the Government shutdown takes place during my permit date, is the mountain off-limits?

While National Parks may close you are primarily on Dept. of Agriculture/USFS land when climbing Whitney (main trail from the Portal). It is not closed. Sheesh, almost the entire state of Nevada would be off limits if all public land were off limits. I would have my permit in hand but it may be less likely to be checked by a ranger.