No Free Rent for Freeloaders Act of 2021 This bill directs the Department of Housing and Urban Development (HUD), on an annual basis, to monitor the extent of noncompliance of public-housing tenants with certain community service and economic self-sufficiency requirements, determine the aggregate amount provided in federal subsidies for all public-housing dwelling units that were occupied by noncompliant tenants, and publish this amount in the Federal Register. In each fiscal year, the amount as determined and published for the preceding fiscal year must be rescinded from funds made available for HUD's Management and Administration account.