What are these dark lines around my mouth and how do I get rid of them? (Photos)

It's difficult to say without the benefit of a formal evaluation. This is my expertise and a huge portion of my practice. Asymmetry is normal and expected. See an expert who does facial shaping with surgical and non surgical methods to get a comprehensive option. Best, Dr. Emer.

Hi,
The dark discoloration is due to increased pigmentation around your lips. This is usually caused by "lip-licking" as it dries out to skin around the mouth when it is done chronically. Also it can happen through consumption of citrus products that stay around the lips and then get exposed to UV light.
The treatment would be to stop the cause of the above. Also some creams prescribed by your doctor can help to accelerate the clearance of it in the meantime. But pay attention to any habits that you might have.
Hope this helped,
Dr. Hamrah @drhamrah

You have peri oral hyperpigmentation which is increased melanin in the skin. Causes are multiple. Treatment would include TCA peel or laser. 

Discolouration around lips, also known as peril hyper-pigmentation can be caused by many reasons, but usually as a result of peri oral dermatitis, which commonly occurs from  dehydrated skin and cold harsh environments. Infection, damage, contact dermatitis and chelitis can be other reasons. For its treatment we can use medical grade chemical peels and skin lightening agents which are very beneficial.
I would suggest you to consult doctor.
Regards
Neel Bector