If it burns to the touch there might be something quite wrong, if your Doctor can`t do anything then... they haven`t done their job right. Try a different Doctor or maybe a Dermatologist.
Itchy dry skin that burns to the touch could be an indicatation for many dermatitis problems. It could be an allergy (have you tried any lotions or body cream, tried new food, been too close to pets, hay fever or new medication? etc.). It could also be a reaction to something mildly toxic (touched any poison ivy, or harsh chemicals?). Or maybe its just extremely dry skin, have you been taking very hot showers and not moisturising after?
As long as you know you`re not allergic, in the mean time you can try numerous different body oils that are very hydrating (not essential oils) Oils such as rosehip oil, argon oil, olive oil, almond oil, grape seed oil, avocado oil, jojoba oil. They are non-comedogenic. Also try Honey, its known to be one of natures magic healers, also same with rose water, but rose water is very light.