Facial burns from superficial chemical peel. Advice?

Without actually seeing you, it is difficult to determine the extent of injury to the skin. If hyperpigmentation (skin that is darker than your normal skin) is present, 2-3 weeks after your peel, consider applying hydroquinone with moisturizer in the morning, and tretinoin cream (retin-a) with moisturizer in the evening. If you are still in the healing phase following the peel, I would apply a thin layer of aquaphor healing ointment (available over-the-counter) daily.
For a light chemical peel, it would be extremely unusual for permanent whitening of the skin to occur. Any darker pigmentation that you have should gradually correct, but will take time (sometimes up to 1 year). A board certified dermatologist or plastic surgeon should be able to differentiate between pigmentation changes and actual scarring that is developing. If you have the latter, you would probably benefit from treatments with a laser e.g. pulsed-dye laser.
Good luck.

Without seeing you or your pictures, it is difficult to answer exactly what happened to you and without knowing the type of peel that was used, it is also difficult to know the depth of the peel that was used and what the end result might look like. Therefore, it is very important to remember that whenever you are having these kinds of procedures performed, that they be performed under the auspices of a board-certified dermatologist or plastic surgeon and that you ask the right questions before the procedure so that you know exactly what they are putting on your skin. In the old days, we have TCA and glycol acids as the most common peeling agents. Now, more and more companies are adding things into propriety peels to make them different from everyone else, and sometimes some of these ingredients can cause a light peel to become more aggressive. If you have a discoloration on one side of the treated area, this probably represents what we call post-inflammatory hyper pigmentation, and over time, this will go away in most everyone. A good skin care regimen is crucial, and this can be determined between you and your dermatologist or plastic surgeon. A sunscreen is also a must. One must also keep in mind that when you put things on your skin, that you can have a reaction to some of the ingredients in those products, so again, someone very knowledgeable in skincare is needed here. You should be fine in the end. Find a good provider and get things back to normal with the proper care

There are several areas of concern with the problem that you are reporting. I am assuming that your chemical peel was done within the last few weeks. It is important to understand what type of peel was done and the time sequences for the problems you are having now.
It is possible to develop a deep burn, like a second degree burn with chemical peel agents. It depends upon what specific agents were used for your peel and how your skin was prepped for the initial procedure.
A TCA peel will irritate the skin causing a stimulation of the pigment producing cells called melanocytes. If you have a hyperpigmentation or darkening it is important that it is properly evaluated and treated. This may include steroids or products such as hydroquinone and kojic acid.
If you actually had a burn then this can lead to scarring and an immediate evaluation is indicated