Remove large pieces of fat from the pork and cut it in to 4 inch pieces.
Make the spice rub and rub it in to the pork pieces.
Place the pork in a glass or metal bowl (plastic will retain the smell of the pork for pretty much the rest of your life), cover with plastic wrap, and refrigerate for at least 1 hour, up to overnight.
After the pork has marinated, place the pieces in the Instant Pot along with the Dr. Pepper and liquid smoke.
Pressure cook on high for 40 minutes with 15 minutes of natural slow release.
Once the pork has finished cooking, remove it from the Instant Pot and lightly shred it in a bowl. When stirring in the bbq sauce later the pork will break down further so no need to shred the pork too finely in this step.
I find there is enough moisture in the pork with the BBQ sauce that I do not need to save the cooking liquid. If you like very saucy pulled pork though, save 1/2 cup of the liquid to put back on the pulled pork at the end. I recommend refrigerating the liquid and skimming off the fat if you do this and can make the time.
Can either refrigerate until ready to serve or add enough BBQ sauce for the pork to be covered and enjoy!
To warm the pork back up, either if from the step above or leftovers, I recommend warming it up slowly, either on the stover or ideally in a slow cooker.