Indian and Alaska Native Foster Care and Adoption Services Amendments of 2005 - Amends part E (Foster Care and Adoption Assistance) of title IV of the Social Security Act to authorize an Indian tribe or intertribal consortium operating a foster care program to receive Federal foster care maintenance payments for children placed in its custody. Requires the State to make such payments if the tribe or consortium does not operate such a program but does have a cooperative agreement with the State for such purpose.