Place pork roast, onion, garlic, and one package of taco seasoning into a large pot. Add water to cover and bring to a boil over medium-high heat. Reduce heat to medium-low and simmer until the pork is tender, checking water level every 45 minutes. Once cooked, transfer pork to a cutting board and reserve 1 cup cooking liquid.

Shred pork using a fork, discarding excess fat. Combine pork, refried beans, diced tomatoes, green chiles, and remaining taco seasoning in a bowl. Add reserved liquid if needed to maintain a moist consistency.

Evenly distribute pork filling down the center of each tortilla. Sprinkle with shredded Cheddar cheese, fold bottom and top edges, and roll to form a burrito.