What is the cause of dark/pigmented lips? and how can it be cured?

The pigmentation on the upper lip and chin may be due to the procedure, threading of upper lip. Since some other material is also used to clean the area before threading, it is possible and likely that it may also be responsible. As you are telling that the skin is dry and pigmented, it means mild irritant type of dermatitis is resulting in post inflammatory pigmentation in those areas. The treatment is to avoid the cause. In your case, threading the upper lip and then use azaleic acid cream available as Aziderm cream (20%) twice daily in the affected areas. Generally, such pigmentations disappears in 3-6 months if such procedures or cosmetics (inducing pigmentation) are avoided.