Ice can be put on a burn for treatment.
But if you do put ice on a burn make sure to put it in some cloth such as a washcloth to prevent the ice from coming into direct contact with the skin which can cause frostbite.
Most times though all you need to do for a burn is to run some cool water over the burn and use some apple cider vinegar or hydrogen peroxide to prevent infection as the burn is healing.
Minor burns only require at home treatment mostly and applying some ice to the burn can help but don't apply the ice directly to the burn without using some washcloth or other material to cover the ice with.
The burn should heal within a few weeks to a month depending on how severe the burn is.
More severe burns will require medical attention but small less severe burns usually heal on their own within a few weeks to a month.