Are there any liability issues with starting a Little Free Library?

As long as you have permission from the property owner and install your library securely and out of the way of foot traffic, you are not likely to have any issues. Be sure to keep your library in good condition (fresh paint, clean, accessible) to encourage people to use it and avoid complaints! Some stewards have decided to purchase personal or commercial liability insurance as a safeguard, but most do not. If you are worried, consult a lawyer for a comprehensive answer.