I have burning nerve pain in my feet. I used to think there was only one type of burning feet pain, but there's really several, pain is much more complex then people say it is.
It depends on the location of your pain- If I were you and you wanted to pursue with a doctor is print out a diagram of different angles of a foot and take a highlighter or colored pencil or marker and draw over where the pain is, if there is multiple types distinguish that. Then ask for an appropriate referal- they'd probably try a podiatrist first, never seen one so can't say how useful they'd be. I assume like most specialists it's very hit or miss. Depending on the type of pain they *might* consider what you are going through as worthy of nerve based inspection, so I'd guess a neurologist... Or if it's very joint focused or allodynia probably a rheumatologist.
My pain is mostly burning but not usually in the soles of my feet (plantar fasciitis), though it is a little more now then it used to be. My pain- toes: felt like someone was stabbing my toes mostly my joints with something jagged constantly; Feet: touching the sheet/blanket hurt (allodynia) - they make 'foot tent' equipment for this; whole foot: and a general burning sensation which is most likely nerve related pain. It might be fibro, it might be peripheral neuropathy which can have several causes - most notably diabetes but that has been ruled out numerous times in my case. I unfortunately can't get these symptoms addressed since other stuff has just been more life limiting, even when these are at their worst.