It is unlawful for a person to possess non-domesticated felines, primates, crocodiles, alligators, and wolves. Their state website does have a permit application available for non-domestic animals though.(Policy Statement by the Department of Game & Fish)
Depending on what category your exotic pet falls into, you may or may not need a license/permit to own them. Inherently dangerous animals as defined by the state are listed as Category 4 animals and require a permit.
Ohio's laws are changing. The most recent laws that have gone into effect have made some pets illegal to own. Things will continue to change until the most recent phase of the recently passed bill goes into effect.