Itchiness and burning after a peel
You do not mention what type of peel you had. It was probably quite mild if you did not experience any flaking. The itchiness and burning may be caused by the products you are now using. You should speak to the person who treated you and ask for recommendations for products. If you continue to have problems, please see a dermatologist.
Burning after a Chemical Peel
It is normal to feel dry and irritated after a chemical peel. You should have been given rich emollients to use after the peel. If you are still irritated after two weeks I would recommend you return to your treating physician for evaluation.