Section 4509.101 of the Ohio Revised Code prohibits a individual from operating a motor vehicle in Ohio without maintaining proof of FR continuously throughout the registration period with respect to that vehicle, or, in the case of a driver who is not the owner, with respect to that driver's operation of that vehicle.
The law requires financial responsibility in the minimum amount of $12,500 for bodily injury to or death of one (1) individual in any one (1) accident, $25,000 for bodily injury to or death of two (2) or more individuals in any one (1) accident, and $7,500 for injury to the property of others in any one (1) accident.
While Ohio has minimum liability requirements (as listed above) they do not have minimum amounts for full coverage. The amount and type of coverage you must place on your leased vehicle will be mandated by your leasing company. The "full coverage" normally required is at least the state minimum liability limits and physical damages of collision and comprehensive.
The Ohio Department of Insurance notes that if your car is financed, the lender will require physical damage protection for the car. If you do not insure the car, the lender may buy the insurance and add the cost to your loan or lease. This is called forced placement. It is expensive and protects the lender, but not you.