The digital rights on the files are date-driven. If the device you're using to read or listen to the overdrive e-book or audio book is set to a date/time past the expiration you'll no longer be able to access the file. I have a Sony Reader and if my library book expires while I'm reading it, I can still finish reading it, but if I back out of the file and try to access from a menu, it catches the expiration and I can't get back into the file.
Now, my Reader has the option for me to modify the current date/time in its settings. As soon as I connect the Reader to my PC via USB, the connection syncs up the date/time setting to match my PC. I would imagine on an iPad or Nook, as the date/time setting would only sync up to current if the device is connected to a server via Wi-Fi, USB, or 3g.