Cut the pork into several large pieces. Mix together all of the dry rub ingredients and coat each piece evenly.
Turn the Instant Pot to the Sauté setting and heat the olive oil. Working in batches, sear the pork until browned on all sides.
Pour the chicken broth, apple cider vinegar, Worcestershire sauce, and liquid smoke into the Instant Pot.
Return the pork to the pot, lock the lid, and cook on High Pressure for 60 minutes. When finished, allow the pressure to release naturally.
Transfer the pork to a large bowl and shred with two forks. Stir in a little of the cooking liquid to keep the meat moist.
Serve plain, topped with homemade vinegar-based barbecue sauce, or tossed with your favorite store-bought barbecue sauce.