You don't say if you're developing a web app, an offline app, or something else. In general, you'll need to use a routing service, or routing code - see, for example, MapQuest's web APIs or Project OSRM's code.
Then do a query for each and compare the distances. There aren't generally services that answer the question "which of these POIs is nearest?" AFAIK, so this is probably the most effective way to find which-is-nearest.