Is it possible for a pelvic infection cause burning in lower back and stomach?

Yes, but uncommon. Pelvic infections can spread to other places, most commonly to the liver (which could cause "stomach" pain, rarely to the kidneys, rarely to the bones. One particular std related to chlamydia can cause reiter's syndrome in which you can get severe arthritis and possibly back pain. If the pain is significant, be sure to rule out the possibility of ectopic pregancy.