I don't know specifically to OH (so treat this as completely ignorant advice) but in general the tax collector gets first dibs on the abandoned home. You might be able to hurry the process along and then bid on it when it is auctioned for back taxes. Or you might be able to strike a deal with the tax collector. You can always say to the tax collector "we won't pay, come levy the taxes and seize the property (p.s. we don't own it anyway)."
In theory. In practice, they can make your life difficult and the payment may be less than the aggro.
You might also be able to follow state-law procedures to correctly notify the tax authority they have such-and-such an amount of time to come collect the home otherwise you get it free-and-clear of any tax liens. That would be according to a state statute that you could look up the procedures for.
You are going to have to clean up those homes anyway if they are abandoned, so you ought to take responsibility now. There is a procedure you will need to follow to take title and the Ohio MHA should know what it is. Then you have to demolish (or if it is less cost, rehab and rent or sell).