Legislative Performance Review Act of 2021 This bill limits the duration of authorizations of appropriations, requires programs with expired authorizations to be terminated, and establishes reporting requirements. The bill prohibits Congress from considering legislation that (1) authorizes appropriations for more than four years, or (2) authorizes the enactment of an unlimited amount of appropriations for any purpose. If Congress determines that an authorization for more than four years is necessary to accomplish the purposes of an authorization, it may waive the time limit by considering a waiver resolution using expedited legislative procedures. Congressional committees that report legislation authorizing appropriations must include in committee reports (1) an analysis, appraisal, and evaluation of existing programs for which appropriations are authorized; and (2) specified details regarding the purpose and need for new programs for which appropriations are authorized. Federal agencies must implement a plan to phase out certain programs with expired authorizations. The bill establishes points of order against legislation that provides appropriations that are either unauthorized or are pursuant to authorizations that violate the time limits established by this bill.