Coronavirus County Relief and Stability Act This bill appropriates $36 billion for a Coronavirus County Relief Fund to make payments to counties to compensate for revenue lost and costs incurred as a result of the COVID-19 (i.e., coronavirus disease 2019) public health emergency. Specifically, a county shall use the funds to compensate for direct loss of revenue due to the public health emergency or to cover its costs that are necessary expenditures incurred due to the public health emergency, were not accounted for in the budget most recently approved as of February 29, 2020, and were incurred from March 1, 2020-December 30, 2020. Each county shall receive a flat payment plus an amount based on its population. The bill provides funding for the Office of Inspector General of the Department of the Treasury to conduct monitoring and oversight of the receipt, disbursement, and use of such funds. The bill reauthorizes the payments in lieu of taxes program, which compensates local governments for tax revenue lost due to tax-exempt federal lands within their boundaries.