Pipeline and Compressor Safety Verification Act of 2021 This bill prohibits the construction and operation of specified natural gas projects until certain Massachusetts state agencies certify that such projects do not pose a danger to surrounding residential communities and the general public. Specifically, the Federal Energy Regulatory Commission must revoke the certificates of public convenience and necessity it issued authorizing the construction and operation of the Weymouth Compressor Station (Atlantic Bridge Project) in Weymouth, Massachusetts, and the West Roxbury Lateral Pipeline (Algonquin Incremental Market Project) in Norfolk and Suffolk Counties, Massachusetts, until it receives written certification from the state agencies.