I haven't tried the below, but I found 2 comments regarding dark skin around mouth.
1: From Anonymous...
"Hi, I too have similar problem...the areas that i have are around the ear, jawline and chest. I was prescribed zinc based ointment, also was asked to apply 'Nizoral' - anti dandruff shampoo, on the affected area, leave it for 15 minutes before washing it. It has been 12 days since i started this treatment....the dryness seems to have reduced and the pigmentation is also turning a shade lighter. Hopefully, it would disappear soon. Apparently, this is some kind of fungal infection which is aggravated by humid conditions. Hope this helps."
"I have had this issue on and off for over 10 years! I am super fair and I still thought I had a moustache because of the discoloration around my mouth! I have had doctors give me many creams and potions and I found that using Selsun Blue shampoo on my spots and as a body wash at least once a week works.
I was in the sun this past week and it all came out again! Humidity is the source of my flare ups! Stay in the air conditioning and try the Selsun Blue. I just bought some! "