Place the pork in a Dutch oven with a lid and season it with the salt, cumin, and oregano. Cover the pork with the onion and the garlic, pour the beer over everything, and set aside.
In a dry skillet, toast the chiles for 3 minutes. Place the chiles in a small pot with 2 cups water, bring to a simmer, and cook for 10 minutes. Transfer the chiles and their cooking liquid to a blender and purée until smooth. Strain the liquid through a fine-mesh strainer over the pork roast.
Cover the Dutch oven with the lid and bake for 2½ hours or until fork-tender. Let rest for 20 minutes, then transfer to a serving platter. Spoon the braising liquid on top and serve with tortillas, lime wedges, red onion, and cilantro.