That is a very unfortunate problem, and it really sounds like it is affecting your quality of life in a big way. Before we even get to some ideas of what could be wrong and what to do about it, it is important to understand that most skin changes need to be looked at by a physician. It is very difficult to describe them in enough detail to diagnose them correctly without a good picture, and there is so much more of the story that a doctor
needs to understand before anything can be done to treat these conditions. First, if the skin redness (erythema, as doctors
call it) looks anything like your eczema that is on the rest of your body, then there is a good chance that it could be eczema and would need appropriate treatment. It would be important to understand if it is your lips that are involved, or your chin/cheeks/etc. While over the counter steroids are often a good choice for eczema on the rest of the body, they should be used VERY sparingly on the face, as the skin is already so thin. Using a steroid on the face can have permanent side effects, and should be done only under the care of a physician. Second, the fact that your skin burns when you sweat could be indicative of some sort of very common fungal infection, although the rest of your story doesn't quite add up. Make an appointment with a dermatologist
as soon as possible. In the interim, try to use a thick over the counter face lotion.