Postresearch Chimpanzee Care Act - Amends title IV of the Public Health Service Act to direct the Secretary of Health and Human Services to provide for the establishment and operation of a national sanctuary system for all surplus federally owned chimpanzees that are no longer needed in research conducted or supported by the National Institutes of Health, the Food and Drug Administration, or other Federal agencies. Mandates that any surplus chimpanzee that is not owned by the Federal Government be accepted into the system if the owner transfers title to the chimpanzee to the Federal Government. Provides for standards for permanent retirement of surplus chimpanzees into the system. Requires such standards to authorize the Secretary to authorize imposition of a fee (except in certain circumstances) for accepting a non-Federal chimpanzee into the system. Outlines provisions for the awarding of contracts for operation of the system. Provides for funding.