Veterans' Heritage Firearms Act of 2013

Veterans' Heritage Firearms Act of 2013 - Provides a 90-day amnesty period during which veterans and their family members can register in the National Firearms Registration and Transfer Record any firearm acquired before October 31, 1968, by a veteran while a member of the Armed Forces stationed outside the continental United States. Grants such an individual limited immunity under the federal criminal code and the Internal Revenue Code with respect to the acquisition, possession, transportation, or alteration of such firearm before or concurrent with such registration. Extends such immunity to a veteran who attempts to register a qualifying firearm outside of the amnesty period if the veteran surrenders the firearm within 30 days after being notified of potential criminal liability for continued possession.

Requires the Attorney General to provide clear notice of, and the Secretary of Veterans Affairs (VA) to carry out an outreach program and develop a communications strategy to provide veterans information regarding, the amnesty and registration period.

Requires the Attorney General to: (1) transfer each firearm qualifying as a curio or relic which has been forfeited to the United States to the first qualified museum that requests it, and (2) publish information identifying each such firearm which is available to be transferred to a museum. Prohibits the Attorney General from destroying any such firearm which has been forfeited until five years after the forfeiture. Requires that any firearm transferred to a qualified museum be registered to the transferee.

Makes a prohibition against transfer or possession of a machine-gun inapplicable to a transfer to or by, or possession by, a museum which is open to the public and incorporated as a not-for-profit corporation under applicable state law.