Terrorist Apprehension and Record Retention Act of 2005 or the TARR Act of 2005 - Amends the Federal criminal code to provide that if the national criminal background check system indicates that a person attempting to purchase a firearm or applying for a State permit to possess, acquire, or carry a firearm is identified as a known or suspected member of a terrorist organization in records maintained by the Department of Justice or the Department of Homeland Security, including the Violent Gang and Terrorist Organization File or records maintained by the Intelligence Community: (1) all information related to the prospective transaction shall be automatically and immediately transmitted to the appropriate Federal and State counterterrorism officials, including the Federal Bureau of Investigation (FBI); (2) the FBI shall coordinate the response; and (3) all records generated in the course of the check that are obtained by Federal and State officials shall be retained for at least ten years.