Hi there. There could be various causes for the burning sensation in the thumb and index finger including neuromuscular and inflammatory conditions, infections, and diseases affecting blood flow to the thumbs. Diseases like diabetes, peripheral neuropathy associated with it can damage one or more nerves causing pain, numbness and burning sensation. Other causes can be traumatic injuries like to the thumb, nerves, burns, herniated discs, nerve entrapment like median nerve, sprains and strains, tendon rupture. Inflammatory cause like arthritis, gout, cellulitis, osteoarthritis, rheumatoid arthritis, etc. neuromuscular causes like carpal tunnel syndrome, raynaud’s disease etc. You need to consult a neurologist for the needful and the treatment will include blood tests, imaging and clinical assessment and therapy depending on diagnosis. Hope this helps. Take care.