This may seem counterintuitive, but certain lip balms can actually make lips feel drier and more chapped after using them over a period of time. They could include irritating ingredients like cinnamon oil, causing contact dermatitis. They could be formulated with only humectants, which, if the air has low humidity, could pull the moisture out of your skin and then it evaporates away.
Or the most likely culprit is that the lip balm contains mineral oil. You see, mineral oil actually has a very large molecular size, making it difficult to impossible to actually penetrate the skin—meaning moisture can’t wiggle its way down into the epidermis.