Specifically referencing "Thor: volume 3", which takes place during the Civil War timeline (which is roughly what timeline the movies currently follow, though they are vastly different in execution). Asgard is brought to hover above Earth, and due to a "disagreement" between Thor and Iron Man regarding the Registration act and the theft of Thor's DNA to create the character Ragnarok, Asgard is declared a Diplomatic Embassy.
This results in an interesting situation, as being a Diplomatic Ambassador doesn't necessarially make you tax exempt. From the research I've conducted, it is primarially based on reciprocity, so "do our ambassadors pay taxes in your country of origin?". Assuming a human of Earth were to form a diplomatic embassy on asgard, it's unlikely that they would be required to pay taxes (as it's basically Norse heaven and likely doesn't have a tax collector) so it can be assumed that as an ambassador of Asgard, he would not have to pay taxes to the foriegn nation of the United States. The question is kind of silly, as j4eo mentioned above, because he doesn't own property or have an income, but it was interesting to research and find out. If you are interested in any of the above info, the state diplomatic information was from U.S.C. 4301-4316, which sets out the rules regarding forigen embasies and their various law (diplomatic immunity) and tax exemptions.