You’ve got that misunderstood. Webmashups like openrouteservice.org and yournavigation.org store (part of) a copy of the OpenStreetMap database in their own format on their mashup server. The slippy map is coming from the OpenStreetMap servers and the route data is queried from the mashup server but drawn on top of the slippy map.

The web mashups read the OSM map data mostly only once a week when the latest OpenStreetMap database dump becomes available.

If a mashup starts generating a significant load on the OpenStreetMap slippy map tile servers then they are supposed to host their own slippy map tiles.