There's also the interesting question of whether he will remain a US citizen after all, or whether taking the office of pope will count as him relinquishing US citizenship under INA §349(a)(4): https://travel.state.gov/content/travel/en/legal/visa-law0/L... In the latter case, the tax question would not arise.
Existing US Department of State policy applies an administrative presumption to most cases of accepting foreign government employment that the person does not intend to relinquish US citizenship unless they affirmatively state otherwise, but they don't apply any such presumption to becoming a foreign head of state or a foreign head of government. They actively analyze such cases individually with no default presumption.
Pope Leo XIV will lose his US citizenship due to his acceptance of the papacy if and only if he intended to relinquish US citizenship by that act, based on the standard of proof of the preponderance of the evidence (the same as in civil lawsuits). He has the right to dispute the question in court if he and the US Department of State disagree on the answer, but I imagine this would in practice be handled more quietly for such a high-profile case.
> A person who is a national of the United States whether by birth or naturalization, shall lose his nationality by voluntarily performing any of the following acts with the intention of relinquishing United States nationality—
It would seem there would need to be an intention to relinquish; which I don't necessarily think is tacit in accepting the office of Pope.
I agree it's not clear that accepting the papacy involves intent to relinquish, but it's not clear either way. The Department of State (at first instance) or any court who considers the matter (if a dispute arises) would normally consider the individual situation in order to conclude what they think is the intent.
In practice, if Trump and Rubio don't want an international incident, they will probably just ask the Vatican what the Pope intended and go along with that.
Two examples from Canada: former Governor General Michaëlle Jean, who represented the Canadian monarch in Canada for day-to-day head of state duties, renounced her French citizenship when before becoming Governor General; and current Canadian Prime Minister Mark Carney renounced his British and Irish citizenships before becoming Prime Minister. Neither renunciation was required according to law or constitutional convention, but they both wanted to remove any question as to their allegiance.