Cut pork loin into 8 portions and sprinkle with salt and pepper.
Turn on a multi-functional pressure cooker (such as Instant Pot®) and select Saute function. Add pork and cook until browned, about 3 minutes per side. Pour in 1/2 cup root beer and cancel Saute function.
Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 20 minutes. Allow 10 to 15 minutes for pressure to build.
While pork is cooking, mix remaining root beer, ketchup, liquid smoke, brown sugar, Worcestershire sauce, grill seasoning, mustard, and browning sauce in a large bowl. Set aside.
Release pressure using the natural-release method according to manufacturer's instructions, 10 to 40 minutes. Unlock and remove the lid. Remove cooked pork and shred with 2 forks and add to the sauce. Serve on toasted buns.