I can't speak for Overdrive as I don't use it. So far on my iPad and iPhone I've only encountered one app that won't run under iOS 7. That app is iExposure and is used for photography calculations. It was never updated for iOS 7 so that is probably why it won't run.

EDIT: I just checked iTunes and iExposure has not been updated in over two years. Another abandoned app I guess. No wonder it won't work.