Congressional Responsibility for Immigration Act - Makes trade authorities procedures of the Bipartisan Trade Promotion Authority Act of 2002 and the Trade Act of 1974 inapplicable to any bill implementing a trade agreement between the United States and any other country if the bill contains any provision relating to U.S. immigration laws or the entry of aliens. Requires the Senate, upon an appropriate point of order, to cease consideration of an implementing bill that violates such requirement. Provides Senate procedures for waivers and appeals of such a point of order.