National Prescribed Fire Act of 2021 This bill directs the Department of Agriculture (USDA) and the Department of the Interior to significantly increase the number and size of prescribed fires conducted on federal lands. Before the end of FY2023, Interior and USDA shall each have carried out a minimum of one prescribed fire on each unit of the National Forest System (with exceptions), unit of the National Wildlife Refuge System, unit of the National Park System, and Bureau of Land Management district (1) that includes an area that has a historical low-severity fire regime, has a historical fire-return interval of not more than 35 years, and is larger than 100 acres; and (2) less than half of which was burned by a wildland fire during the previous 10-year period. Interior shall establish a collaborative prescribed fire program to provide financial assistance to eligible entities to conduct prescribed fires in priority landscapes. Interior shall establish an incentive program to encourage the implementation of large, cross-boundary prescribed fires. Interior and USDA shall hire additional employees and provide training to increase the number of skilled and qualified prescribed fire practitioners in the departments and other organizations. To address the public health and safety risk of the expanded use of prescribed fire, USDA and Interior shall conduct research to improve or develop, among other things, cost-effective, and consistent mitigation strategies for communities impacted adversely by smoke from prescribed fire.