Summarize:

It's best if the person who is bleeding applies pressure to his own lip, but if you must assist, be sure to wear clean rubber gloves. Using a clean towel or piece of gauze or a bandage, apply gentle but firm pressure to the cut for a full 15 minutes. If the towel, gauze or bandage becomes completely saturated with blood, apply additional gauze or bandages without removing the first layer. The cut may trickle or spot blood for upwards of 45 minutes, but if there is steady bleeding after the first 15 minutes you may need to seek medical help.  The mouth—including gums, tongue, and lips—has a lot of blood vessels and a heavy blood supply, so oral lesions tend to bleed more than cuts on other parts of the body.  Apply the pressure inward, toward the teeth, jaw, or gums. If this is uncomfortable for the injured person, place gauze or clean cloth between the person’s teeth and lip, then resume applying pressure. If the bleeding has not stopped after 15 minutes of steady pressure, if the injured person has problems breathing or swallowing, if he has loose teeth or if his teeth seem out of their usual position, if you are unable to remove all dirt or debris, or you are concerned he may have other injuries to his face, you should contact a doctor to see if the injury needs stitches or other professional treatment. Do this as soon as possible, as the chances of infection increase the longer you leave a wound open and bleeding. If you have any doubt, make sure to contact a doctor.  If the cut goes all the way through the lip, it's important to seek medical assistance immediately. If the cut is on the red part of the lip as well as on the normal colored skin above or below the lip (crosses the vermillion border), the injured person should see a doctor for stitches. Stitches will reduce the risk of infection and help make sure the wound heals in the best cosmetic way. Doctors recommend stitches if the cut is deep and gaping, meaning that you can place fingers on either side of the cut and gently pry it open with minimal effort.  Doctors may also recommend stitches if there is a flap of skin that can easily be sutured.  Deep lacerations that require stitches should not wait more than 8 hours, maximum, to receive safe treatment.
Apply pressure. Check the wound after 15 minutes. Contact a medical professional if necessary.