I think the "Little Free Library" locations might be similar to a "public bookcase" http://en.wikipedia.org/wiki/Public_bookcase
https://en.wikipedia.org/wiki/Public_bookcase
There was a discussion thread on the tagging mail list a while back that might pertain to this at https://lists.openstreetmap.org/pipermail/tagging/2013-August/014401.html
I don't see that any consensus was reached on that but two possibilities did come out of that:
1. Tagging it as a reuse facility: http://wiki.openstreetmap.org/wiki/Reuse where the tags would be something like amenity=reuse, reuse:books=yes
2. And simply amenity=public_bookcase
I prefer the second single tag variety but haven't committed to it for tagging a bookshelf/library on a stand by a church that is a mile or two from me.