How Can I Treat Very Dark Lip Corners and Hydroquinone Does Not Work? (photo)

From your photo, it looks like the darkening of the lip corners is due to post-inflammatory hyperpigmentation from eczema or irritation.  People with braces, for example, can get such darkening from irritation of saliva.  This can also happen if products used for dental hygiene have alcohol or very strong flavoring, like mint or cinnamon.  This is usually called angular cheilitis and needs to be treated with prescription anti-inflammatory and anti-yeast medications (not hydroquinone).  Please consult a board-certified dermatologist for treatment.

I would also suggest that putting a little petroleum jelly in the lip corners at night time will protect them and help fade away the dark patches and to discontinue any oral hygiene products that have alcohol or strong flavors.

Make sure that you are not licking the angles of your lips on a regular basis. Also use sunscreen in that area. There is a product called Elure that may help, but it only treats the problem (if it works) and not not cure it.