For a similar task, I have used a combination of Osmosis and Osmfilter:
1. download an extract for the desired area (e.g. http://download.geofabrik.de/north-america/us/new-york.html )
2. find out the coordinate box you're interested in (the above gives you the entire Empire State :))
3. extract a .osm file (substitute the box values from step 2, the below ones are obviously nowhere near NYC):
`osmosis --read-pbf file=inputfile.pbf --dataset-bounding-box bottom=49.75 top=50.35 left=14.05 right=14.87 --write-xml file=nyc.osm`
4 . filter the resulting XML using osmfilter:
`osmfilter nyc.osm --keep="amenity=" --keep-ways="amenity=" --keep-relations="amenity=" > nychospitals.xml`
amenities.xml`
The above will give you everything tagged "amenity" - this should cover many POIs; check the OSM wiki for other tags which you may want to see.