Burning and tingling in the feet are most frequently due to a peripheral neuropathy. This is a length dependent process which affects the parts of the nerves furthest away from the body (i.e. the toes followed by the feet). There are many causes to neuropathy such as diabetes, rheumatologic disease, vitamin deficiencies, blood disorders (i.e. paraproteinemias such as nmltiple myeloma or other monoclonal gammapathies), kidney failure and infectious processes such as syphilis. There are also genetic disorders that cause peripheral neuropathy, the most common being Charcot Marie Tooth disease.
Sometimes nerve roots impingements of the back can mimic a peripheral neuropathy, particularly S1 nerve root impingements.