Responsibility for Unaccompanied Minors Act This bill amends provisions relating to responsibilities of the Department of Health and Human Services (HHS) for the care of unaccompanied alien children. The Office of Refugee Resettlement of HHS may provide care for an unaccompanied alien child until the conclusion of the child's immigration proceedings or the child reaching 18 years of age, whichever occurs first. Before placing an unaccompanied alien child with a sponsor, the office shall evaluate the child's future need for follow-up services. Before placing an accompanied alien child, HHS shall (1) require the proposed sponsor to execute a sponsor care agreement, (2) notify the welfare agency of the state where the child will be placed, and (3) conduct a background check of the sponsor and all adults living in the household in which the child will reside. Under the care agreement, each sponsor shall agree to (1) provide for the child's physical and mental well-being, and (2) ensure the child's presence at immigration proceedings and compliance with all court and agency orders. HHS shall assume physical custody of a child if the non-parent or legal guardian sponsor fails to comply with the care agreement. HHS shall periodically report to Congress on instances where a sponsor failed to comply with a care agreement. The Department of Justice shall increase the number of immigration judge teams by at least 225.