It's day 3 after a chemical peel, my skin is burning. Will it leave permanent marks?

Hi it will depende the type of compound to determine the deep of the damage . and the color of your skin is something to consider too.

It really depends on the depth of the chemical peel (superficial, medium, deep).  Petroleum jelly is always good to heal superficial wounds but you should really check with your physician's office where the peel was performed as they will know what the typical downtime is for the specific peel you had done.  A superficial peel typically peels on day 2-4 and then recovers quickly whereas a medium or deep peel can take a week or more to really recover.