As I've said elsewhere, the UN will not be considered a prince, king or a foreign power or a state other than the US. The UN is an international organization founded by the governments for the benefit of the government. Face it, there were no such organizations in the days of Washington.
Furthermore, as evidence proves, this particular statement in the constitution is, as they would say here, not a "living statute;" that is, it has become obsolete by continuous practice to the contrary. In no country is the constitutional law an exact discipline, not here, not in the US. While this can be seen as nefarious or cynical, the "living constitution" is the practice common to all the world, and it is a necessary evil, since no document can last forever against the merciless march of time. Constitutional law is somewhere in between hard law and soft law, that is, it is binding, but it is possible to overrule a statute by continuous practice.
If we were to take these statements in the constitution as hard law, it would preclude all citizens of the US from becoming officials in international organizations like the Red Cross or Amnesty International lest they lose their citizenship. This was, obviously, not the intent of the legislator (here the founding fathers), and it is absurd.
For this reason, and because of the aforementioned status of the UN as the supra-governmental organization founded by the governments for the governments (and thus is not in the conflict of interest with the good of the US), I expect that if this ever becomes a court case, this will most likely be one of the argument why not to impeach Obama.