A white coating with burning pain on the tongue for this long usually points to oral thrush, a yeast overgrowth, or if nothing has cleared it so far, something like geographic tongue or burning mouth syndrome. Since you've already ruled out
HIV and
diabetes, those are the likelier possibilities left.
Next Steps
See a general physician or dentist for a tongue scraping or swab test to confirm which one it is. An antifungal mouth rinse is often tried first if thrush turns out to be the cause.
Health Tips
See a doctor sooner if the white patches spread, you have trouble swallowing, or you lose weight without trying.