I have to throw in a correction to the answer above. There is no device I am aware of that allows the eReader app to simply remove the book when the loan has expired. The eBook will show some kind of indication that the loan period is expired and if the user tries to open it they will get some form of error message that tells them they no longer have access to the book.
All the eBook apps that I'm aware of do not remove the eBook from your device because of security concerns. If an app has the power to erase something from a device, then it presents an exploit that could be used to damage data on the device. So to be on the safe side, the app makers do not have that kind of access built into their software. The apps can download and store, but they cannot remove. (At least not automatically, the user has to initiate the deletion.)
There's another service that kind of competes with OverDrive called Freading. We use it here at my library system and it's another method of providing eBooks for our patrons. Freading is different from OverDrive in several ways but the big two are the fact that Freading doesn't really have much from the big six publishers and how Freading does downloads.
See, with Freading, you get an allotment of tokens per week. The tokens are virtual of course but they allow you to download a limit of books per week. Here, we have four tokens per patron. Each eBook is a set number of tokens. Some are one, some are four, but you get the idea. You can download one four-token book (checks out for two weeks just like OverDrive) or you could download four one-token books. The library is paying Freading by the download so, unlike OverDrive, the eBooks are always available. No holds, no waiting, no out of stock like you see with OverDrive which has a model that's much more like a "traditional" library collection.
Freading uses Adobe Digital Editions for authentication and they have apps for iOS and Android. However any eReader app using ADE will work with Freading too and even Freading recommends the BlueFire reader for iOS or Android. Like OverDrive, it's a two week check out and the item simply will not open after the loan period has expired. The user can remove the eBook from their device, but the app will not do it for them.
Another difference between OverDrive and Freading is the renewal process. While neither of these services has a renewal mechanism built in, the patron can simply re-download the book from Freading at a whim if they didn't finish. With OverDrive, if the book is popular, and there's a waiting list, then that person will have to wait to finish it.