What is the best method to remove a flat facial mole from upper lip?

For the lip, I believe that moles respond really well to excision and closure. This would be done surgically but under local anesthesia and typically in the office. You would be trading the mole for a straight scar but most people are very happy with this trade off- you need to determine whether you find this acceptable. Other methods- laser, burning, freezing, shaving- can remove the mole but will do so incompletely and you risk it coming back with a greater likelihood than excising it. 
Not everyone is a good candidate for this treatment so please see a Board Certified or Board Eligible Plastic Surgeon to evaluate if you may benefit. Good luck!
This content is for educational and informational purposes only. It is not a substitute for medical advice, diagnosis, or treatment provided by a qualified healthcare professional.  You should always seek the advice of your physician or healthcare professional for any questions you have about your own medical condition. 

Your options in general include some type of shaving with cautery or surgical removal- its hard to say unless you post a picture. That will be more helpful. 

This needs an in-person evaluation. I recommend following up with your cosmetic dermatologist. Moles can be removed with laser, radiowave or shave. See an expert. Best, Dr. Emer

Fortunately, there are multiple options for removing a mole from the upper lip, and typically either involve a laser or surgical excision.  No matter the approach, it's impossible to remove a mole without some kind of scar, and it's up to you what kind of scar you'd rather have.  In my opinion, surgically removing a mole is best, because it gives the most control, and most predictable result, that is, a straight line scar.  Other techniques, like laser, or a shave, are less predictable in their outcome.
A visit to a plastic surgeon's office will help you to understand if you're a candidate for the procedure, and what aesthetic outcomes are realistic in your particular case.  In order to get a qualified, ethical, and expert opinion on your surgical options and expectations, always schedule an in-person office consultation with a plastic surgeon certified by the American Board of Plastic Surgery.
Best of luck,
Keith M. Blechman, MD
New York, NY

Mole removal is a little bit more complicated in terms of the mole's color, whether the mole is in a crease or line as well as whether the mole is raised or flat.  Treatment options including various lasers, such as an Alex laser or CO2 or erbium laser, shave removal using a special technique, or a simple surgical excision of them mole.