I doubt a fast can eat up loose skin, it is healthy normal tissue...but your skin is highly adaptable whether fasting or not and it takes truly massive weight loss to get 'loose skin' - weightloss unlikely to be achieved in a single fast (30 lbs or something is not going to produce loose skin I should think).
One reason though people appear to have "loose skin" is because they actually have lost weight in such a way where they've lost a lot of connective tissue and muscle tissue before losing the bulk of their subcutaneous fat, so the fat and skin hang off the body in thick rolls (thicker than actual skin by itself). This shouldn't happen during even an extended fast since you mostly lose fat not muscle and connective tissue.