Things may be different in Ohio, but Property Tax is generally levied on the market value of the entire property (including structures). I suspect that most of the value of the property is in the house, so even if you could give that section away I don't think it will change the overall value much.
That said, why would the city take that land from you? Is there some value to the city that makes it worth their effort? There's no standard procedure for giving land back to the government, so it would have to be a willing transaction. It seems like it's locked in by your property and other wooded areas. The only thing that gives me hope from the picture is the nearby baseball fields.
Instead of giving it away, I would talk to the owners of adjacent lots and see if that land has value to them that would make it worthwhile to buy it from you cheaply.