If he desired to attain U.S. citizenship, I doubt that losing his status as a Counsellor of State would be that much of a deterrent (it's highly improbable that HRH The Duke of Sussex will be called to play that role), nor would having to forsake other governmental entities including his father as head of state beyond personal feelings (there are many people with dual U.S./U.K. citizenship). I doubt that HRH The Duke of Sussex would run for American public office where he would need to relinquish all of his foreign titles, but who knows? We'll cross that bridge if we get to it.
As a naturalized citizen, he would have by law to renounce all of his foreign titles prior to naturalization. Curiously, a natural-born citizen does not have to renounce foreign titles (although he/she cannot accept one while holding a public office without Congressional consent ), but naturalized citizens do.
I don’t think the renunciation would actually have any legal effect in the United Kingdom though as British law does not allow people to disclaim a peerage except within one year after
succeeding to one, which does not apply to Harry.
In any case, it would be weird for Harry to continue calling himself Duke of Sussex after renouncing the title. And, given that their whole brand ( Sussex.com, the coat of arms, etc. ) is built around him and his wife being called Prince Harry, Duke of Sussex, and Meghan, the Duchess of Sussex, I don’t see Harry giving that up to become an American citizen.
Separate answer (unrelated to the comment above): Here is
US State Department guidance on reliquinshing or keeping U.S. citizenship when accepting a position in a foreign government. Apparently, a review has been ordered on existing policy, so I cannot guarantee that the information on the link (from 2022) is still valid.
