From Catholic News Agency:
His election as pope raises several interesting questions from a nationality law perspective. Chiefly: can a U.S. citizen become king of a foreign country and still remain an American citizen? Paul Hunker, an American immigration attorney and a Catholic, told CNA that U.S. federal law — specifically 8 U.S. Code § 1481 — spells out some very specific conditions under which a U.S. citizen can lose his or her citizenship. These can include committing an act of treason, obtaining naturalization in a foreign state, and, crucially, accepting a position as a foreign head of state. The key, though, is that in order to be “expatriating,” these things must be done by a person voluntarily and with the intention of relinquishing their U.S. nationality.