There are several solutions in the web to check that:
http://www.openrouteservice.org/ http://www.yournavigation.org/ http://maps.cloudmade.com/ http://osmu.org/demo/index.html?lat=52.33062&lon=5.62056&zoom=16&layers=B000FTFTTT&v=motorcar&adj=recommended&markers=!52.33013%2C5.62329!52.33299%2C5.62199!52.32888%2C5.61748 http://open.mapquest.com/ (not working currently) http://routingdemo.geofabrik.de/ (http://project-osrm.org/)
I’m currently using a lot the Openstreetmap map for navigation in my android application Navdroyd to navigate from one point to another. Just remember that turn restriction can take a while till they are applied to the routing. weeks , month