I'm going through the same thing. It started off as Angular Cheilitis and after applying all of the creams to get rid of it, I'm left with eczema around and on my lips. I saw on you tube that a girl that suffered from eczema stayed taking shots of extra pure virgin olive oil. 3 tablespoons to be exact. I would suggest putting all 3 tablespoons in a cup and drinking it at once and having a glass of water or juice right after. Doing table spoon by table spoon is hard after the first.
I'm also drinking Chinese herbal tea to detox, eczema Is a way of your body trying to get rid of something toxic.
Clean your face with a soap free cleanser, I use Soy face cleanser by Fresh. I dab my lips dry with a clean paper towel and apply lubriderm, which stings at first or eucerin.