The Twilight Zone - Season 3 Episode 8 Our Selena is Dying

A young woman, Diane, comes to live with her family, including the dying patriarch Sylena. Much to the doctor's astonishment, Sylena begins to make an astonishing recovery, while Diane begins to show all the signs of rapid aging. The doctor soon deduces the truth: the Brockman's have discovered a way to assure immortality, at the sacrifice of their younger members. When the doctor confronts Sylena, a fire breaks out burning down the house. Diane is restored to her normal youth, and Sylena is taken in as a Jane Doe, and begins to heal herself as a nurse nearby complains of inexplicable burn injuries.