Trim the excess fat off the pork, leaving a thin layer on top. Poke all over with a knife; place in a large bowl. Puree 10 of the garlic cloves with the beer in a blender. Pour over the pork, cover with plastic wrap, and refrigerate overnight.
Heat oven to 325° F. Place the pork fat-side up in a roasting pan. Add the beer marinade. Sprinkle 2 tablespoons salt and 2 teaspoons pepper all over the meat. Cover with foil and bake until forktender, about 4 hours.
Meanwhile, grate the chili and the remaining garlic clove on the small holes of a box grater set over a bowl, then grate the tomatoes and onion on the large holes. Stir in the lime juice, honey, and 1 teaspoon salt. Refrigerate for at least 1 hour.
Uncover the pork. Spoon off the excess fat. Heat broiler. Broil the pork until the top is crisp, about 6 minutes. Cool for 20 minutes.
Shred the pork and toss with its juices. Serve with the buns and salsa.
* Total Time includes marinating