What can I do about hyperpigmentation after a chemical peel?

It is very rare to have burns after superficial chemical peels. Discoloration can happen as a result of not applying sunscreen or not following the post-procedure instructions.
Avoiding any exfoliating products before the peel is also very important. When the skin is hypopigmented, it takes up to six months for some of the pigment to restore in the skin.
Hyperpigmentation can be treated with hydroquinone, or a bleaching cream, but your doctor needs to decide if you are ready for that. Conservative IPL treatment or a mild microdermabrasion can also help with any brown pigment left from a chemical peel.

A 30% glycolic peel with a microdermabrasion is still a relatively superficial treatment. However, it is still possible to have a post-inflammatory hyperpigmentation. As other physicians have noted, there are many options for dealing with this issue. It is also not uncommon for the hyperpigmentation to resolve on its own.
However, I guess I am left wondering if you lack confidence in your treating physician. Has there been enough break down in communication that your trust with this doctor is shot? It might be a good time to seek out another physician to assist you if this is the case. This does not mean that the first doctor 'botched" the job. It really means that for whatever reason you have lost enough confidence that it is a good idea to seek help elsewhere.

Chemical peels are effective in the treatment of aging skin including reduction of pigmentation, fine wrinkles, and acne scars. Depending on skin type and complexion, patients may require pre and post-procedural treatment for hyperpigmentation. Generally, Fitzpatrick skin type and ethnicity are factors to consider prior to performing a chemical peel.
If you are experiencing excessive pigmentation after a chemical peel, you should consider speaking to your physician about a post-procedural depigmentation protocol to control the extent and severity of skin pigmentation. This may include a course of topical skin modulation with retinoids and/or hydroquinone.

With all chemical peels there is the risk of post inflamatory hyperpigmentation, which means blotchy, dark spots. They occur when the skin gets irritated from too much rubbing makeup off or scrubbing off the exfoliating skin too early or sun exposure during the first 2 weeks when your skin is especially sensitive.
Regardless of how or why it occurred, it is fixable with either further lighter chemical peels to exfoliate the dark spots since they are usually superficial or prescription bleaching creams to bleach out the dark spots.
I like to use Jessner's solution peels or 10% TCA peels every 2 weeks in addition to having patients apply a triple bleach cream every other night to start with and eventually nightly sparingly. And of course, daily sunscreen use to protect from the sun.
After each peel, I have patients use a prescription cortisone cream or lotion, such as desowen, twice a day to prevent inflammation or irritation and take away redness faster. Go back and continue your treatments to correct the dark spots but do expect to pay for each further peel or creams, since it is a normal risk you, yourself, have to take with all chemical peels.

Some darker skin types have more sensitive or fragile skin. A side effect of this condition, is that in the presence of heat (like a chemical peel), more pigment is produced. A treatment for this condition is Finacea and some mild 2% lactic acid peels. The pigmentation will lessen over time.