Why is it that when I wear high heels my foot starts to burn?

Pain. The position of the foot/ankle complex is in when wearing high heels is not a normal position. There are abnormal pressures put on the bones, muscles and nerves of the calf, ankle and foot. The pressures and strain on the nerve causes pain which, in your case, is perceived as burning. Burning pain is a type of neuropathy. So, the wearing of high heels is causing you to have neuropathic symptoms.