Is rain water heterogeneous or homogeneous?
Assuming we isolate rain water and only rain water, it is homogeneous.
A homogeneous mixture cannot be separated into its individual components via mere filtration.
Rain water qua rain water (rain water taken as itself) is a homogeneous mixture.
Rain water, as a mixture isolated from random particulates on the ground, has components that cannot be further filtrated out: nitrates, nitrates, phosphates, chlorides, bromides, and other ions that aren't water.
(As a sidenote, these ions can be identified via an Ion Chromatography experiment; I've done this with beer before. :D)