Tinnitus Research and Treatment Act of 2013 - Directs the Secretary of Veterans Affairs to: (1) recognize tinnitus as a condition for research and treatment by Department of Veterans Affairs (VA) Auditory Centers of Excellence; (2) recognize the need to enhance the research and treatment of tinnitus and to ensure the allocation of appropriate resources for such research and treatment conducted by such Centers, commensurate with peer-review protocols; and (3) ensure VA cooperation with the Hearing Center of Excellence established by the Department of Defense (DOD) to further research on tinnitus.