What can happen if you have several missed court dates in another state?

Almost certainly, there is an Ohio warrant out for your arrest, which warrant can be enforced by Illinois law enforcement. ASAP you need to contact a skilled Ohio criminal defense attorney located near the courthouse where your case was at, to get the case resolved. Plus, you need to make arrangements ASAP w/a family member, or a friend, etc., someone you trust, for them to be able to bail you out of jail if you call them if you're arrested, which could happen at any time.