The United States was founded by the act of expatriation of citizens from England, but ironically the U.S. itself actually did not grant its own citizens the right to renounce citizenship.
"There is no right to expatriation in either the Declaration of Independence, the Constitution or early federal laws."
Until the the mid-19th century, the United States implicitly followed the English common-law tradition of perpetual allegiance, a feudal concept in which all natural-born citizens are considered to inherit upon birth a debt of obligation to the country in which they are born for the state protection they receive.
Because this feudal debt of obligation can never be canceled, the citizen can never relinquish his citizenship.
The irony of the U.S. following a system of "perpetual allegiance" was not unnoticed by lawmakers at the time. On the one hand, the U.S. welcomed and protected immigrants who expatriated from their countries. On the other hand, the U.S. itself continued to follow a medieval concept prohibiting expatriation.
In 1868, Congress passed a law to rectify the situation and explictly endorsed "the right of expatriation".
"Praising expatriation as a natural and inherent right of all people, Congress declared in 1868 that any act of the government which denies, restricts, impairs, or questions the right of expatriation is inconsistent with the fundamental principles of this government."
The language in this law clearly recognizes the "right to expatriate". It's important to note, though, that this act does not create a "right" in the legal sense. It's not a right similar to free speech, practice of religion, etc. It's not written into the Constitution. And although it can't be known with certainty until tested, it seems fairly likely that Congress could revoke this "right" at any time and permanently ban renunciation of citizenship. The Constitution gives Congress the power to establish rules of granting citizenship (naturalization), so they presumably also have the implicit power to determine the rules of taking it away (expatriation). (If you're interested, see our discussion here for a more detailed analysis of the power of Congress to change or rescind the right to renounce U.S. citizenship).
Expatriation can take two forms: the right of the citizen to sever his ties with the state, and the ability of the state of sever its ties with the citizen. Through several laws and court decisions, both sides of expatriation have been modified.
At various times in the early 20th century, it was held that an American could lose citizenship through acts such as military desertion, failure to pay taxes, acceptance of citizenship of a foreign country, and even a woman's marriage to a foreigner (the Supreme Court ruled in 1915 that an American woman who marries a foreigner loses her U.S. citizenship while the marriage remains in effect, but would resume citizenship upon the dissolution of marriage).
Finally, in the case of Afroyim v. Rusk in 1967, the U.S. Supreme Court overturned these rulings, declaring that involuntary loss of citizenship was not constitutional and that nationality can be taken away from a U.S. citizen only if he gives his consent. A later Supreme Court ruling in 1980 (Vance v. Terrazas) and the subsequent Immigration and Nationality Act passed by Congress in 1986 further clarified that this consent to relinquish citizenship cannot be implied through acts such as voting in a foreign election; rather, there must be a clear preponderance of evidence that the citizen intended to renounce his citizenship.
"It is now settled law that every U.S. citizen may renounce citizenship at any time. On the other hand, the government may not withdraw nationality from a citizen based on any actions the citizen takes; loss of citizenship can only result when the there is clear evidence that the person intends to expatriate."
Since 1990, the Department of State policy based on these laws and court rulings has been to revoke an individual's citizenship if and only if an individual appears in person at a U.S. embassy or consulate abroad and explicitly declares his intention to renounce citizenship.