Mass Evacuation Exercise Assistance Act of 2005 - Amends the Robert T. Stafford Disaster Relief and Emergency Assistance Act to provide for grants for mass evacuation exercises for urban and suburban areas and for the execution of emergency response plans. Requires each state or local government receiving such a grant to develop detailed and comprehensive mass evacuation plans for each area in its respective jurisdiction. Requires the Secretary of Homeland Security to assist states and local governments in developing and maintaining such plans.