What are the treatments for healing a burnt esophagus ?

Burned Esophagus: It depends what burned it... Your own stomach acid is quite different than lye ingestion. Assuming it's from your own acid (gerd) you could take Prilosec otc but i would also consult with your doctor... Sometimes it's a treatable infection (h. Pylori) or a more serious issue that needs attention from a GI physician.

Rest and time: Not sure exactly what you mean by burnt, but the only burn an esophagus can suffer is a caustic injury. This can be life threatening and require serious surgery and rehab. If the burn is not full thickness then some rest and nutrition and time are needed.