Really cool idea, will suggest it for my local municipality! Maybe the library could be interested, they usually have to cull books regularly due to lack of space.
In the very small town I live in, the library did just that. They started out with selling books they were trying to get rid of, but also offered trade in of books you had finished with yourself. Now the trade-in shelves are well-used. The only rule is that you are not supposed to bring back a book that you got there. Other than the fact they accumulate a huge number of John Grisham novels, it is great! (I don't have anything against JG novels, but I'd like to read something else too.)
At my current contract in the city, my office has a library that has a bookcase for books and magazines too. There is no trade-in requirement, though most people do. Again, well used. I take books there that can't be re-sold. Always exciting when someone in the building decides to declutter! We have had an issue with a group of romance-readers who read enormous numbers of romance novels and trade them among themselves and then bring them to this bookcase, causing a bit of a glut. This week I took in a supply of other books -- thereby taking romance out of the library and putting a little mystery into it.
I like the Givebox idea. Our recycle centres have something like that, but you have to drive out to the landfill where it is located.