What causes constant leg and arm muscles feel like they are on fire

Lots of things: Many things can do this. If both limbs on the same side are burning, you could have had a stroke, or less likely a spinal cord problem. If all 4 limbs are involved, then the possibilities are actually quite more numerous but also include brain stem stroke & spinal cord disorders. Other causes: myositis, polyneuropathy, arthritis, lupus. For answers you need an examination & other tests.