For how long does it take for a swelling from a burn to subside?

As long as needed. Burns are the response of the tissues to injury from heat. It is simply inflammation which means increase blood flow to the area to promote healing. Depth determines care. Do not use ice worsens already injured skin. Cool water soaks 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.