Transient burning feet:
(1) sun or thermal burn
(2) skin irritant exposure (poison ivy, contact allergen, etc), including shoes
Persistent burning feet:
(1) polyneuropathy (diabetic, small muscle fiber neuropathy)
(2) erythromelalgia secondary to polycythemia (including chronic mountain sickness)
(3) malabsorption with vitamin deficiencies (celiac disease, postoperative, etc)
(4) autosomal dominant familial burning feet (hereditary sensory neuropathy)
(5) spinal neurenteric cyst or arteriovenous malformation
(6) toxin exposure (rodenticide RH-787 = Vaccor B, ergotism)
(7) tarsal tunnel syndrome
(8) metatarsalgia
(9) plantar fasciitis
(10) podoconiosis
The presence of both burning hands and feet tends to favor a polyneuropathy, including chronic mountain sickness.
An older adult may have both a polyneuropathy and a structural problem in the foot causing a compression neuropathy.