Burma Unified through Rigorous Military Accountability Act of 2021 or the BURMA Act of 2021 This bill imposes sanctions pertaining to Burma (Myanmar) and addresses related issues. The President must impose property- and visa-blocking sanctions on certain foreign persons (i.e., an individual or entity), including those that (1) knowingly operate in Burma's defense sector, (2) are responsible for or complicit in undermining Burma's democratic processes, or (3) are senior leaders in Burma's military or government. The Department of the Treasury must prohibit or impose strict conditions on certain accounts used to facilitate transactions for such sanctioned persons. The President may impose sanctions on Myanma Oil and Gas Enterprise if such sanctions would support certain objectives, including reducing the Burmese military's ability to undermine democracy in Burma. The President may also prohibit imports of precious and semiprecious gemstones from Burma. Before removing certain foreign persons from a list of specially designated nationals and blocked persons (commonly known as the SDN list), the President must certify to Congress that the person in question has not knowingly engaged in certain activities, such as supporting terrorism. The bill authorizes Department of State and U.S. Agency for International Development activities in Burma and the surrounding region to support democracy activists, humanitarian assistance, and reconciliation efforts. The State Department may (1) continue to assist organizations supporting political prisoners in Burma, and (2) provide assistance to entities investigating crimes against humanity. The President must direct U.S. representatives to the United Nations to vote and advocate for certain actions related to Burma, such as cutting off assistance to Burma's government.