Burma Unified through Rigorous Military Accountability Act of 2019 or the BURMA Act of 2019 This act imposes sanctions, authorizes assistance, and requires the establishment of various programs and strategies related to Burma (Myanmar). The President shall impose sanctions against current or former officials who are responsible for or who supported serious human rights abuses in Burma, as well as against entities controlled by such officials and military-linked entities that provide such officials significant financial benefits. The sanctions shall include asset blocking and visa blocking. The bill authorizes humanitarian aid for Burma, Bangladesh, and the surrounding region for various purposes, including aid for ethnic minorities targeted by Burma's military and support for voluntary resettlement of displaced persons. The Department of State shall maintain a publicly available website listing entities in Burma's gemstone sector that meet certain transparency and due-diligence requirements related to ownership, control, and supply-chain standards. The State Department shall also issue guidance for U.S. companies as to best practices for supply-chain due diligence when dealing with Burmese gemstone entities. The State Department shall report to Congress on topics including (1) accountability for war crimes in Burma, and (2) a strategy for promoting economic development in Burma.