With little information, I can only somewhat guess what could be happening. Most moisturizers will not do that crack and peel effect unless the elements beforehand are somewhat abrasive (toners, larger granules in exfoliating washes) or it may not be moisturizing enough. Too much fragrance in a moisturizer over burned and raw skin can cause painful dry patches too. The first thing to try and solve the problem would most likely be to change the cleanser.
Cleanser suggestions to try:
REN - Since your profile says combination, Rosa Centifolia cleansing Gel. If that sounds too drying, then the cleansing milk
Skyn Iceland -Glacial Face Wash (at their official site), Pure Cloud Cleanser
Boscia - Soothing Cleansing Cream
Caudalie -Gentle Cleanser
More than likely once you get a good gentle cleanser that works for you, the moisturizer should not be too drying or burn the skin. If it does, check to see if it contains fragrance or menthol and look for ones that do not have either ingredient.
Try to prevent using your hands or shoulders as resting places for the chin. Also try not to rest the chin onto hot surfaces (cell phones heat up). Do not to expose the chin out into the colder elements for too long.
If you do use a toner in your skincare routine, skip out or find one that is alcohol free. That should help prevent the burning. Also do not over exfoliate the area as that will cause the skin to feel dry and raw. If the area seems a bit inflamed or starts developing questionable sores; schedule an appointment with a physician or dermatologist.
Best of luck!