Biological Weapons Policy Act of 2021 This bill modifies the Department of State's authorities related to biological and virological weapons and addresses related issues. The State Department's authority to engage in certain activities (e.g., conduct research and provide grants) related to detecting and reducing certain weapons shall include activities directed at biological and virological weapons. The bill prohibits using federal funds for research that increases the pathogenicity, contagiousness, or transmissibility of viruses or bacteria if the research involves a foreign entity subject to the jurisdiction of certain countries, including China and Russia. The President must carry out certain activities related to China's compliance with the Biological Weapons Convention, including engaging with other governments about concerns about such compliance. The U.S. representative to the United Nations (UN) must advocate to bar representatives from certain countries, including China and Russia, from serving in leadership positions in any UN agency or program with a nexus to certain subjects, such as global health and biosecurity. The U.S. mission to certain countries must complete an assessment of whether each U.S. government-supported project involving life sciences dual-use research of concern aligns with U.S. national interests. The Government Accountability Office must report to Congress on (1) U.S. government authorities and processes related to cooperation with other nations in life sciences research that could be weaponized or pose dual-use concerns; and (2) coordination between federal agencies responsible for public health, biomedical research, and national security in the assessment of the security implications of cooperation with other nations on life sciences research.