No, dual from birth doesn't count as a relinquishment. There has to be intention to lose citizenship. You can't do that if you're dual from birth.
Here are the ways you can relinquish/renounce your citizenship:
Potentially Expatriating Acts
Section 349 of the INA (8 U.S.C. 1481), as amended, states that U.S. nationals are subject to loss of nationality if they perform certain specified acts voluntarily and with the intention to relinquish U.S. nationality. Briefly stated, these acts include:
1.obtaining naturalization in a foreign state upon one's own application after the age of 18 (Sec. 349 (a) (1) INA);
2.taking an oath, affirmation or other formal declaration of allegiance to a foreign state or its political subdivisions after the age of 18 (Sec. 349 (a) (2) INA);
3.entering or serving in the armed forces of a foreign state engaged in hostilities against the United States or serving as a commissioned or non-commissioned officer in the armed forces of a foreign state (Sec. 349 (a) (3) INA);
4.accepting employment with a foreign government after the age of 18 if (a) one has the nationality of that foreign state or (b) an oath or declaration of allegiance is required in accepting the position (Sec. 349 (a) (4) INA);
5.formally renouncing U.S. nationality before a U.S. diplomatic or consular officer outside the United States (sec. 349 (a) (5) INA);
6.formally renouncing U.S. nationality within the United States (The Department of Homeland Security is responsible for implementing this section of the law) (Sec. 349 (a) (6) INA);
7.conviction for an act of treason against the Government of the United States or for attempting to force to overthrow the Government of the United States (Sec. 349 (a) (7) INA). https://travel.state.gov/content/tra...tionality.html
Number 5 is the renunciation, the rest are considered relinqishments. If done without the intent to lose citizenship then it isn't lost, but if you do it with intent then that is grounds for applying for a relinquishment. Some people, when applying for a second citizenship will prepare a document for the person overseeing their oath taking to sign to say they did it with the intent to lose their US citizenship.
For a relinquishment you'd be asked to fill in the following form to help the embassy/consulate staff decide whether or not you have done so. Unfortunately, it's more designed to help those people who may have committed an expatriating act without intending to lose their citizenship rather than for those who do intend to so. https://eforms.state.gov/editdocumen...documentid=240
Here is an example from Isaac Brock Society's website of the intent to relinquish by gaining another citizenship: http://isaacbrocksociety.ca/2011/12/...ce-if-you-can/
But unless you've done one or more of the other 6 possibilities you can't relinquish, you can only renounce.