Franklin Eddy said:
Probably not at this time. Libraries are not carrying e-books in Kindle format yet. Amazon said that Libraries should start carrying books in Kindle format later this year.
Of course, if the Kindle software is upgraded to read epub books then it becomes easier.
Actually I'm not sure ePub will make it easier - library lending will be more streamlined with Kindle if the books come in .mobi/azn format, because that would eliminate the need for using Adobe Digital Editions the way we do with ePubs. No one (outside of Amazon & Overdrive) knows for sure how library lending will work - it's been speculated that the books could come directly from Amazon's servers much like they do with Kindle-to-Kindle lending - but we don't really know yet. The fact that bookmarks, notes & highlights made on the library edition would be saved if one chose to purchase the book helps support that opinion, though.
But no, since current Overdrive books are almost always ePubs they can't be read on a Kindle without jumping through some hoops involving DRM removal, which we can't discuss here. But they'll work on tablets or computers with the proper apps (Overdrive, ADE, or for iPhone/Touch/iPad, the Bluefire Reader app ).