Will Sun Exposure After Chemical Peel Cause Permanent Tanning?

Chemical peels leave your skin sensitive and vulnerable to damage. During recovery, treating your face with the proper skin care products and wearing protective sunscreen will ensure you do not hamper your results and prolong your recovery. Sun exposure is responsible for much of the damage that chemical peels work to eliminate, so prolonged exposure to the sun immediately following a chemical peel, even with sunscreen, is not recommended.
Your best bet to fix the damage is to continue following your doctor’s advice. Without photos or more knowledge of your skin’s condition, making a different recommendation than what your doctor has prescribed would be difficult and ill-advised. If you have any further questions, be sure to talk them over with your doctor. Hope this helps.

Sounds like your Dermatologist is doing just the right thing with a combination of a bleaching agent (hydroquinone) and a topical corticosteroid.
You need to be cautious about sun exposure soon after a peel to avoid prolonged redness which can then lead to post-inflammatory hyperpigmentation, and if severe, (but unusual with a medium peel), permanent scarring.

Be cautious with any further sun exposure.
It typically takes somewhere between 3-4 months for this to resolve. Some areas of hyperpigmentation may take longer (up to two years).

Hello,
After a medium strength or deeper peel, I keep patients out of the sun for several months. Deliberate tanning after a peel can cause pigment shifts that may take more peeling to correct particularly in darker skinned people. Follow your surgeon's advice carefully to get through this.

Sun exposure will cause skin damage, and work against what the peel is trying to accomplish. Eary exposure to sun after skin resurfacing can cause permanent hyperpigmentation, and therefore should be avoided until the area is completely healed. People with deep coloration to the skin such as hispanics, asians, etc. are particularly at risk for this problem.
You need to avoid further sun exposure until cleared by your doctor. Skin bleachers like Triluma are excellent. Only an evaluation of your current status can tell for sure what you should be on, as it will change as you heal.

My suggestion would be to discontinue mometasone at this time. Cortisone creams should not be used long-term on the face (or anywhere else for that matter). I would encourage strict sun protection/avoidance, using a sunscreen containing zinc oxide and/or titanium dioxide, SPF 30 or greater, daily.
Tretinoin cream can be applied over moisturizer in the evening, and hydroquinone over moisturizer in the morning. If irritation/redness occur, you should use these topicals less frequently (e.g. every other day or alternating days).
Good luck.