This is not true... If you go and do some research you will find that you must cool the burn first with cold water, not ice, cold running water and this is only for 1st and 2nd degree burns. 3rd degree burns, keep them breathing, cover with wet clothes but DO NOT immerse in water.
Egg whites can contain salmonella and can cause infection and even death. If you insist on using them as mentioned in one medical reference in the late 1890s... cool the burn with water first...
Placing egg whites on a burn is dangerous. Cold water will cool a burn quick. Egg whites will hold the heat in and continue the burning. It can also cause an infection if the skin is broken. Run the burn under cold water and seek medical attention.