Heat 2 tablespoons oil in a 4-quart heavy pot over moderately high heat until hot but not smoking, then sauté onion, stirring occasionally, until golden brown, about 6 minutes. Add carrots and celery and cook, stirring, until softened, about 5 minutes. Add lentils, water, salt, and pepper and bring to a boil, then reduce heat to moderate and cook, covered, until lentils are tender, about 30 minutes.
While lentils are cooking, heat remaining tablespoon oil in a 10-inch nonstick skillet over high heat until hot but not smoking, then brown sausage on all sides, about 5 minutes. Cool sausage slightly on a cutting board, then cut into 1/2-inch-thick pieces. Add sausage to soup and stir in vinegar.
