Health Insurance for the Unemployed Act of 1991 - Amends the Federal-State Extended Unemployment Compensation Act of 1970 to entitle individuals receiving extended benefits under the unemployment compensation program as trade readjustment allowances to benefits under Medicare's Hospital Insurance program (part A of title XVIII of the Social Security Act). Allows such individuals also to elect family coverage under such program. Requires individuals electing coverage under such program to pay a weekly premium to be deducted from their unemployment benefits. Requires the amounts deducted for such premiums to be transferred to the trust funds supporting the Medicare program.

Outlines provisions concerning: (1) the identification by the Secretary of Labor (the Secretary) of the individuals electing such coverage and the weeks for which they are entitled to health insurance benefits, for transmission to the Secretary of Health and Human Services; (2) termination of health benefits eligibility; (3) agreements between the Secretary and a State for administering the health insurance benefits program for the unemployed established by this Act; (4) administration of such program absent a State agreement; (5) penalties in connection with the disclosure of false information to receive benefits under such program; and (6) regulations to carry out such program.

Amends the Hospital Insurance program to make such program benefits available on the same basis as provided for other individuals entitled to Medicare Hospital Insurance program benefits, except that enrollment in a Medicare cost or risk-contracting health maintenance organization would not be permitted.

Amends the Internal Revenue Code to allow the unemployed individuals who have elected Medicare Hospital Insurance coverage under the Federal-State Extended Unemployment Compensation Act of 1970 to suspend COBRA continuation coverage.