This famously happened to Boris Johnson a while back. He was born in New York so was a US citizen (no longer). He was denied boarding going to Mexico for a family holiday via Houston because he didn't have a valid US passport. He ended up rerouting so he didn't go through the US.
Ah yes in 2006. He then said he would renounce his US citizenship and was so adamant that he would do so that he even renewed his passport in 2012!
He then did renounce but was only to after settling a US tax bill on the sale of a house in London.