What can I do about severe sunburn swelling of the face around the eyes?

Cool compresses. Sunburns are just like heat burns - they cause swelling. Cooling the skin with ice can lessen the swelling. If the skin is blistering, your doctor may prescribe antibiotic ointment. If the swelling does not go down in a day, see your doctor.
Simple Care. Sunburn is response of the tissues to injury from solar radiation. It is simply inflammation which means increase blood flow to the area to promote healing. Do not use ice as this can injure already injured skin. Cool water soaks can help as can taking anti-inflammation medicines like Motrin or advil (ibuprofen). Avoid any further damage as this can cause scarring . Good skin cream applied daily/gently helps.