Health Insurance Tax Relief Act of 2005 - Amends the Internal Revenue Code to: (1) allow individual taxpayers a refundable tax credit for health insurance costs paid for the benefit of the taxpayer, the taxpayer's spouse, and dependents; (2) require business taxpayers who receive payments for certain employee health insurance coverage to file informational returns; and (3) direct the Secretary of the Treasury to make advance payments of health insurance tax credit amounts to health insurance providers.