Comprehensive TB Elimination Act of 2021 This bill reauthorizes for FY2022-FY2027 and revises the tuberculosis (TB) prevention and treatment grant program administered by the Centers for Disease Control and Prevention. The Health Resources and Services Administration may also award grants to state, tribal, territorial, and local governments and specified health care providers (e.g., Native Hawaiian health organizations and federally qualified health centers) for TB prevention and treatment efforts. Additionally, the Government Accountability Office must report on (1) national TB prevention and treatment efforts, and (2) matters concerning TB drug shortages and costs.