According to OverDrive:
"When an existing OverDrive Media Console (OMC) user upgrades to iOS 7, the app will fail when trying to open a DRM-protected eBook.
"New or first time users of OMC are unaffected by this issue meaning that if you download the app for the first time post-iOS7 update you will not experience this issue.
"We are resolving the issue but in the meantime, the immediate remedy for users is to uninstall and re-install OMC which will also require the user to re-authorize with Adobe.
"IMPORTANT NOTE: A re-install will clear a user’s bookshelf, history, and app settings."
See more at: http://overdriveblogs.com/