Brown turkey in a skillet over medium heat until no longer pink, 5 to 7 minutes. Season with salt and pepper; drain and discard any excess grease. Add black beans and diced tomatoes; cook until warm, about 5 minutes. Reduce heat to low.
Preheat the oven to 300 degrees F (150 degrees C).
Warm cooking oil in a skillet over low heat. Heat tortillas 1 at a time, about 1 minute each. Fill with turkey mixture, and top with cheese. Place burritos on a baking sheet.
Bake in the preheated oven until cheese melts, about 5 minutes.