Tiahrt Restrictions Repeal Act This bill amends several appropriations laws to remove limitations on the authority of the Bureau of Alcohol, Tobacco, Firearms and Explosives to conduct activities related to the administration of federal firearms laws. Specifically, the bill removes provisions that: limit the use of firearms tracing data, prohibit the imposition of a requirement that firearms dealers conduct a physical inventory, require national instant criminal background check records to be destroyed within 24 hours.