Will Results from a Chemical Peel Be Permanent?

Thank you for your question; because you have a tendency for eczema I would recommend speaking with a dermatologist before getting any procedures. Treating inflamed skin can only make the discoloration worse and you run the risk of creating a bigger problem. Depending on the severity of the discoloration and your skin tone you may benefit from laser treatments to remove the pigmentation. Without seeing the areas it is hard to recommend the best treatments, your dermatologist will be able to give you different options for services.
-Best wishes

First of all, it sounds like you need to manage this problem with a board-certified dermatologist--especially given your history of eczema.  Do not do chemical peels on yourself.  As far as whether the effects of chemical peels are permanent, the answer depends on what one is treating.  If one treats hyperpigmentation from acne and eczema with a chemical peel and the primary skin problem is under control the effects should be "permanent."  If one is treating a chronic problems such as melasma or aging, the effects will likely not be.

You should consult a dermatologist to examine your skin and offer appropriate treatment for you.  Do not self treat with peels.