Money Follows the Person Act of 2005 - Authorizes the Secretary of Health and Human Services to award grants to states for demonstration projects (MFP demonstration projects) designed to: (1) increase the use of home and community-based, rather than institutional, long-term care services (rebalancing) under state Medicaid programs; (2) eliminate barriers or mechanisms that prevent or restrict the flexible use of Medicaid funds to enable Medicaid-eligible individuals to receive support for appropriate and necessary long-term care services in the settings of their choice; (3) increase the ability of state Medicaid programs to assure continued provision of home and community-based long term care services to eligible individuals who choose to transition from an institutional to a community setting; and (4) ensure that procedures are in place to provide quality assurance for eligible individuals receiving Medicaid home and community-based long-term care services and to provide for continuous quality improvement in such services.