I have yet to see an example of eczema of the lips that stays exclusively on the mucous membrane and is not spreading on the surrounding skin of the lips. And if you have cracks and redness on the corners of the lips, that is a sign of angular cheilitis.
I feel like a broken record sometimes recommending against topical treatments. But my experience with Aquaphor white petrolatum (admittedly during a time of Depression and stress) leads me to believe this is most likely a counter-productive treatment only to be used for acute cases of dry lips. I wish I could have avoided the long term effects -- I think I set the healing process back a couple years.
Though I don't know your specific case history, if you're like me and have had this for 5 years, applying nothing and easily peeling off a layer of hard, thin skin every 7 days (that's not noticeable except the last couple days) is much more preferable to slightly swollen lips and soggy, flappy, white skin that peels every 2-3 days. Please note that diet and stress levels have changed drastically between those times for me.