Twenty-First Century Dams Act This bill addresses dam infrastructure and safety. First, the bill provides resources for programs related to state dam safety, the rehabilitation of high hazard potential dams, and the inspection of dams. In particular, the bill reauthorizes through FY2026 the National Dam Safety Program. In addition, the bill reauthorizes through FY2026 a program related to the rehabilitation of structural measures and a program related to water infrastructure. Next, the bill allows a new tax credit for maintaining and enhancing hydroelectric dams that is equal to 30% of the basis of any qualified dam safety, environmental, and grid resilience enhancement property placed in service in a taxable year. It also allows a 30% credit for qualified obsolete river obstruction removal expenditures for demolishing and removing certain nonpowered or hydroelectric dams. The bill directs the Department of the Army to establish a program to carry out dam removal projects. Further, the bill establishes the Dam Removal Council to (1) coordinate with participating federal agencies to annually notify all known dam owners of the availability of dam removal funding and provide guidance on the existing programs of participating agencies; (2) evaluate and submit to Army recommendations for dam removal projects, technical assistance programs, and funding allocations; and (3) develop a dam removal strategy. The bill requires certain dam assessments, including a joint assessment by the U.S. Geological Survey and the Department of Energy. The bill also authorizes specified federal agencies to undertake certain activities related to federal dam infrastructure.