Promoting Health for Future Generations Act of 2014 -  Amends the Internal Revenue Code, with respect to health savings accounts (HSAs), to: (1) increase the contribution limitation for such accounts, (2) allow Medicare beneficiaries and participants in health care programs administered by the Veterans Administration (VA) to enroll in and contribute to such accounts, and (3) allow adult children of deceased holders of Archer Medical Savings Accounts and HSAs to receive tax-free rollovers of remaining amounts in such accounts.