The complaint of “burning feet” immediately makes me think of peripheral neuropathy but let me clarify. Peripheral neuropathy is a disease of the nerve itself. This disorder is fiber length dependent. That means that the disease normally affects the longest nerves in the body first. The S1 and L5 nerves are the longest nerves and these supply the feet.
The nerve itself becomes “sick”- possibly from losing its blood supply (the vasonervorum) and starts to deliver ectopic messages. That is- the nerves produces signals for no reason. These signals typically are of a “burning” nature and are symmetrical (both feet equally). The symptoms typically increase at night and some patients complain of hyperalgesia (pain with just the covers touching the feet). Yes- diabetes is associated with this disorder but many patients have no apparent associated diseases that can cause peripheral neuropathy.
A disc herniation at L5-S1 can cause these symptoms but this is highly unlikely. Both S1 nerves would have to have equal compression for symmetrical symptoms and it is extremely unusual for the symptoms not to start in the buttocks and radiate to the feet.
Seek out a good neurologist as these specialists can diagnose this disorder with an electrical test and there are medications that can treat the symptoms.