North Korea Sanctions and Diplomatic Nonrecognition Act of 2013 - Continues diplomatic, economic, and military sanctions against the government of North Korea as a supporter of international terrorism until the President certifies to Congress that North Korea: (1) is no longer engaged in the illegal transfer of missile or nuclear technology, particularly to Iran, Syria, or Burma; (2) is not assisting foreign terrorist organizations, engaged in counterfeiting U.S. currency, or engaged in illicit narcotics traffic; (3) has returned the last remains of Reverend Kim Dong-shik to the United States; (4) has released abducted Japanese nationals and surviving Korean War prisoners of war; and (5) has undertaken specified actions regarding family reunification, penal reforms, refugee access, and religious organization communications.

Expresses the sense of Congress that the Secretary of State should redesignate North Korea as a state sponsor of terrorism.

Finds that the United States did not grant diplomatic recognition to North Korea upon its establishment as a client regime of the former Soviet Union in 1948.

Continues diplomatic nonrecognition of North Korea until such benchmarks have been met.

Directs the President, in the case of a North Korean missile, rocket, or other airborne launch or the conducting of a nuclear test in violation of U.N. Security Council Resolutions 1695, 1718, and 1874, to instruct the U.S. Permanent Representative to the United Nations to use U.S. influence to secure adoption of a Security Council resolution condemning North Korea's action and requiring implementation of comprehensive sanctions and an inspection regime against North Korea.