You wouldn't have burning from HPV, nor would you likely have symptoms from HPV that soon. The incubation period - time from infection to symptoms - is usually several weeks to months for HPV.
Have you seen a doctor? You weren't at any risk for an STD, unless there was some actual grinding and you were both naked from the waist down, but perhaps you have something else going on. If you mean that your bum is burning internally, you could have some inflammation in your prostate.
It's been going on for 6 months. It's time to see a doctor.