First of all, we need to find out what is causing your lip sweating. Are you taking any drugs? Are you eating a lot of spicy foods and hot food? Are you allergic to anything? If so, then just stop consuming that particular food or drug. You can also ask for help from someone who knows better than me; perhaps the doctor or pharmacist would know more about this matter.
If none of those things are true (and I really hope they aren’t), then let’s see how we can cure your upper lip sweating problem with natural remedies: eat healthy and avoid foods that cause excessive heat in the body such as dairy products, red meat and alcohol. Make sure that you drink plenty of water and take some herbal supplements like papaya leaf extract (to protect mucus membranes) and rosemary essential oil (for detoxifying). Also make sure that your diet contains lots of vegetables which will help alkalize your body!