What would cause a sensation that your sock is wet in your shoe when it's not?

Neuropathy: Which is basically altered sensation. See a doc to find out the etiology of this problem.

Neurological disease: This sensation is a form of a dysesthesia or paresthesia, an abnormal sensation. The cause of which can be from a problem with the nerves themselves (like in a peripheral neuropathy) or a central nervous condition (brain related).