I had liposuction on my lower back and fat transferred to my buttocks 3mo ago why do I still have burning sensation in my lower back?

Nerve recovery. One of the most common, usually completely reversible consequences of liposuction is change in sensation. Areas of numbness are expected after liposuction due to minor trauma to sensory nerves but complete sensation is expected to return within days, weeks or at most, months. This symptom can be minimally bothersome or more so. Discuss your symptoms with your surgeon for advice.
Self limiting. Liposuction is used to removed fat from target areas without significant damage to nerves and blood vessels. It is common to have some dysesthesias [burning] following liposuction and fat transfer due to nerve injury, irritation. Trauma induced edema of the tissues and the nerves should slowly resolved. As the swelling and inflammation resolve, so should the dyesthesias.
Probably normal. It is very common to have some dysesthesias (pins and needles) following liposuction and can occasionally be more pronounced following ultrasonic assisted liposuction. Generally this will , resolve with time. Topical massage and desensitization exercise may help to diminish the discomfort.
Nerve Irritaion. It sounds like you may be experiencing irritation of the local nerves after liposuction surgery. This may be related to localized injury and/or inflammation. In either case, these sensations tend to resolve with time, without treatment. Discuss your concerns/symptoms with your plastic surgeon. Best wishes.