No. Ohio's radon law, ORC Chapter 3723, licenses testers and mitigators but does not require landlords to test rentals. Local health districts such as Delaware's note they have no statewide authority over indoor air in homes. Tenants can test their own unit with a kit.
More detail
For an owner, the practical case for testing is simple: ground-contact units carry the exposure, results are cheap to get, and a high number is better found by the owner than by a tenant. ODH recommends retesting multifamily buildings every five years and houses every two. Mitigating a rental works like any home, done by an Ohio-licensed mitigation contractor, with the closeout package kept for the next tenant or buyer. A tenant who finds a high result should send the report to the owner in writing; free kits are available through ohio.radon.com. For obligations under a specific lease or a local housing code, ask a lawyer rather than relying on general guidance. See radon in apartments and condos.