The mouth and jaw are controlled by the Trigeminal Nerve V, and the inside of the lower mouth under the tongue is controlled by the Hypoglossal Nerve XII. A lower lip droop may be an indication that the nerves to the mouth, the jaw and the muscles below the tongue might have been affected.
One should find out what has caused the lower lip to droop. Has there been a minor CVA, i.e., cerebrovascular accident? Did one recently go to the dentist and the dentist used an overdose of anesthetic? Is there any sensation in the drooping lower lip? Is the lower lip drooping and the gum of the bottom row of teeth is showing? Is there a loss of feeling of the skin around the lower cheek and the lower jaw? Are there any sores inside the mouth that would have affected the Hypoglossal Nerve XII? Is there any kind of lower gum infection? Did the dentist apply anesthesia at the right place and with the right dosage? Has the lower jaw been hit in perhaps a boxing match and the lower jaw has receded?