I had a lumbar strain or sprain 3 months ago, but still have pain. when will it fully heal?

Should have healed: The natural history of low back pain is that it gets better within a month or so. If it hasn't please see your doctor, or a specialist in physical medicine (physiatrist), osteopathic physician who does manipulation or a trusted chiropractor. Core and abdominal strengthening programs are very helpful. Make sure your seating and work ergonomics are good, and eat an anti-inflammatory diet.

It is more than a : Strain as it would have been healed by 6-8 weeks later. You need to get evaluated as to what is the true diagnosis unless it is improving otherwise. Then, it will most likely continue to resolve even if a disc is involved 90% of the time but it could be a few months.