PostGIS : Does the <-> operator work for Geography?

No, the <-> operator is not defined on geography. You can use the geometry operator and a cast to geography on the resultants

ST_Distance(geom::geography, ST_GeogFromText(''))

to get the final distances, but potentially if your objects are over the poles or dateline or in the far north the initial geometry ordering of <-> won't be correct so things won't work perfectly.

You can use ST_DWithin with geography in PostGIS

--Find the nearest hospital to each school
--that is within 3000 units of the school.
-- We do an ST_DWithin search to utilize indexes to limit our search list
-- that the non-indexable ST_Distance needs to process
--If the units of the spatial reference is meters then units would be meters
SELECT DISTINCT ON (s.gid) s.gid, s.school_name, s.the_geom, h.hospital_name
    FROM schools s
        LEFT JOIN hospitals h ON ST_DWithin(s.the_geom, h.the_geom, 3000)
    ORDER BY s.gid, ST_Distance(s.the_geom, h.the_geom);

--The schools with no close hospitals
--Find all schools with no hospital within 3000 units
--away from the school.  Units is in units of spatial ref (e.g. meters, feet, degrees)
SELECT s.gid, s.school_name
    FROM schools s
        LEFT JOIN hospitals h ON ST_DWithin(s.the_geom, h.the_geom, 3000)
    WHERE h.gid IS NULL;

http://postgis.net/docs/ST_DWithin.html

You can find more information about the logical operators of postgis in another question:

Clarification of Logical Operators in a PostGIS Database