I did locate this information which maybe helpful for new Kindle Fire owners.
Go to our eMedia Catalog an get the Overdrive app from overdrive.com/software/omc. To get the app, you need to click on Mobile/Android device and then click where it says “If your device doesn’t support Android Market, you can download OverDrive Media Console for Android from OverDrive.”
Once you have an item checked out and downloaded via the Amazon app (that part works the same as the Kindle eReaders downloads from our eMedia Catalog), go to Settings/Wi-Fi and tap on Sync. Then the book will show in the bookshelf and be able to be read.
The other problem with the app as it is now, unlike the app for iPad that only allows you to checkout and download material that will work with the iPad, the Kindle Fire app allows the user to checkout anything – some formats will not work on the Kindle Fire but are still considered checked out to the user (taking up some of their limit) and they are not returnable via the Amazon login. Kindle Fire only works with Kindle editions and MP3 audiobooks. No video, music of WMA formats.
Supposedly it will download epub editions but it requires the user to have an Adobe ID. Really hoping Amazon continues their cooperation with Overdrive and allows a better app to be made available.
Other comments about the Kindle Fire and Library eBooks can be found at the Amazon.com Customer Discussion site.