You have a couple issues happening:
Many shaving creams have zero moisturizers. That will leave your lips dry and chapped. Try Noxzema shaving cream and see if it makes a difference.
Shaving by its very nature is an exfoliating process. You're removing the top several layers of skin cells. Your lips don't really have those cells to spare, and may be trying to heal. Be extra careful shaving around your lips. I actually press my lips together, trying to roll the soft fleshy parts into my mouth, leaving only my skin exposed to be shaved.
Women DO have this exact same issue when trying to shave their, umm, bikini areas. That's why many choose to wax instead. That and the fear of nics and cuts.