In a crock pot, combine the tomato sauce, taco seasoning, and green chiles; stir. Add the chicken to the crock pot and turn to coat in the sauce. Turn the crock pot to low and cook 6-8 hours.
Preheat oven to 400 F.
Remove chicken from crock pot and shred between two fork; transfer to a large bowl and top with 3-4 tablespoons sauce from the crock pot. Reserve remaining sauce in the crock pot and turn crock pot to high.
Cook the rice according to package directions. Once cooked and hot, stir in the lime juice and cilantro; keep warm.
In each tortilla, add ⅓ cup rice and about 5 ounces of shredded chicken. Roll the burritos, folding in the sides to contain the filling. Place the burritos seam side down on a baking sheet. Bake for 5-10 minutes or until tops begin to brown.
Top each burrito with about ¼ cup reserved sauce from the crock pot, a few tablespoons of cheese, tomato, and lettuce. Serve immediately.