As a last resort, ice or cold compresses can be used to treat inflammation, or to reduce burning or itching. We say last resort because cold compresses, or ice, on the eyes can cause meibum, which should flow easily, to harden. In a hardened state, like cold butter, meibum will not flow. However, for acute cases, or for short periods, cold compresses or ice are fine to use.
You can also try applying ice to an area near the site where you feel the burning. By applying ice to a nearby location, e.g.: your upper cheek or eye brow, you will take focus away from the burning sensation in your eye. See Tight Shoes for more information about redirecting focus to alleviate symptoms.