Burma Human Rights and Democracy Act of 2013 - Prohibits FY2014 Department of Defense (DOD) funds from being made available to the government of Burma unless the Secretary of State certifies to Congress that: (1) Burma has taken steps toward establishing civilian oversight of the armed forces, addressing human rights abuses by the military, and terminating military relations with North Korea; (2) Burma has taken steps to establish a fair and inclusive process to amend the Constitution of Burma, including the full participation of the political opposition and ethnic minority groups; (3) Burma has amended its constitution and laws to ensure civilian control of the military; (4) Burma is promoting peace agreements or political reconciliation and is addressing the resettlement and humanitarian situation of displaced persons; and (5) the Burmese military is improving its human rights record, taking steps to withdraw forces from conflict zones, and implementing a code of conduct.

Directs the Secretary of Defense to report annually to Congress on the status of military-to-military engagement between the U.S. Armed Forces and the Burmese military.