Veterans Air Travel Streamlining Act of 2014 - Requires: (1) the Administrator of the Transportation Security Administration (TSA) to waive the application fee for veterans with a service-connected disability rated at 50% or more who apply to participate in TSA's Pre-check program, and (2) the Administrator and the Secretary of Veterans Affairs (VA) to establish a system to verify such a veteran's disability rating and whether such veteran's disability is service-connected.