The specific pain from a lip piercing will vary person to person based on different factors. Piercings are painful, regardless of where on the body they are, as piercing creates a puncture wound.
Many of those who have personal experience claim that the lip piercing itself does not hurt. Factors such as the thickness of the skin in the location of the piercing and whether or not there is scar tissue in the area will determine the level of pain experienced.
Typically, if a piercing has to go through scar tissue, it will be more painful. The needle itself going through is normally not what hurts most; after the needle comes the insertion of the ring or stud, which are generally thicker than a needle. When the jewelry is pushed into the new piercing, the hole stretches and this is typically the most painful part.
A couple of hours after the piercing, swelling will start, which can also be painful. Holding a small ice cube in the mouth against the lip will help with the pain, and crushing a baby aspirin and mixing it with a drop of water will make a paste that you can pack on the outside of the lip to bring down swelling. An important factor to consider is the length of the jewelry. If you choose a stud, make sure it is long enough to accommodate swelling; if you choose a ring, make sure the circumference is large enough to fit the initial swelling.