There's a historical backgrounder here.

This part stands out:

Federal law and regulations establish procedures, administered by the Department of State, by which Americans can voluntarily renounce their citizenship. In addition, federal law lists a variety of acts that shall result in the loss of citizenship if "voluntarily perform[ed] . . . with the intention of relinquishing United States nationality." These include obtaining naturalization in a foreign state; declaring allegiance to a foreign state; serving in the armed forces of a foreign state as an officer or when the foreign state is engaged in hostilities against the United States; and, in some cases, serving in governmental office in a foreign state.

Of course, there are millions of U.S. citizens who are also citizens of other countries. And, recall the case of Illinois state senator Martin Sandoval, who wants to also serve on a Mexican committee.