Housing for the Chronically Mentally Ill Act of 1986 - Amends the Housing and Community Development Act of 1974 to permit State mental health authorities, or other private nonprofit or public agencies designated by a State, to use vacant residential property to provide supervised living for the chronically mentally ill. Authorizes appropriations for FY 1986 and for each fiscal year thereafter. Amends the Housing Act of 1959 to set aside specified amounts beginning in FY 1986 for handicapped family housing loans. Amends the United States Housing Act of 1937 to authorize a housing voucher program for the chronically mentally ill. Authorizes FY 1986 through 1988 appropriations.