My aunt had something like this, which I believe was misdiagnosed several times before receiving an accurate diagnosis. I believe her diagnosis was trigeminal neuralgia. Her symptoms included excruciating neurological pain and burning at times, and was misdiagnosed as having to do with her teeth at first. Once she was properly diagnosed, she had an operation done and the pain was completely gone. I don't know if there are enough similarities to suggest that you have the same thing, but it sounds like I'd be visiting a neurologist if it continues, and perhaps even a couple of them if one doesn't have much to say. She had to visit multiple before getting the right diagnosis and eventually the right treatment.
I agree - I was thinking that the whole time I was reading your post. I really hope you get some answers. Let us know.