Chapped lips never feel good. While moisturising them with lip balm will go a long way towards making them feel better, you also need to get rid of the dead skin. Using a toothbrush on your lips can remove it.
To remove the dead skin from your lips, apply a bit of lip balm or Vaseline to your toothbrush and brush your lips. Obviously, you’ll want to use a separate brush for your lips or clean it thoroughly before you use it for regular tooth brushing.