Combine salt, paprika, garlic powder, onion powder, and black pepper in a bowl. Cut chicken into 3- to 4-inch pieces; place in a large resealable plastic bag. Sprinkle salt mixture over chicken, squeeze out excess air, seal the bag, and shake to coat chicken.

Pour water into a multi-functional pressure cooker (such as Instant Pot). Place the wire rack into the Pot; arrange chicken on the rack. Close and lock the lid. Select Meat/Poultry function. Set the timer for 50 minutes. Allow 10 to 15 minutes for pressure to build.

Release pressure using the natural-release method according to manufacturer's instructions, about 10 minutes.

Transfer chicken to a bowl; shred using 2 forks. Add 1/2 cup cooking liquid and liquid smoke to the bowl; mix well. Serve pulled chicken on buns; top with barbecue sauce.