I can't speak to requirements for the UK monarchy, but on the US side, it appears to be legally possible for someone in line to the throne to become President. The only constitutional requirements are:
No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty five Years, and been fourteen Years a Resident within the United States. [ref]
No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once. [ref]
Obviously, there's no guarantee of being elected, and the nature of the situation practically writes the attack ads from opponents (similar to criticisms of Kennedy about being a Catholic and thus answerable to the Pope).
However, it's a very different story when it comes to actually inheriting the crown while in office. Article I, Section 9, clause 8 of the Constitution (also called the Emoluments Clause) states
No Title of Nobility shall be granted by the United States: And no Person holding any Office of Profit or Trust under them, shall, without the Consent of the Congress, accept of any present, Emolument, Office, or Title, of any kind whatever, from any King, Prince, or foreign State. [ref]
I think everyone can agree that becoming the King or Queen of the UK would constitute "accepting an Office or Title from a foreign State". Thus, the POTUS/heir would require Congressional approval to accept the crown. Without it, they could choose to resign as POTUS or abdicate the crown, but they could not hold both.
The same logic may apply to being named Crown Prince/ess (due to the death or abdication of a prior claimant). The person in question would still be the heir apparent, but would be unable to receive any of the associated lands or titles until Congress approved or they left office.
It's also worth noting that none of these requirements are unique to the President. Anyone holding any elected or appointed office in the US would have the same restrictions on taking the crown without Congressional approval.