Can driving with overdrive off damage the transmission?

No, Turning the overdrive off will only cause the rpms of your vehicle to be higher. Typically you would turn overdrive off when you need more power and aren't worried about your fuel economy. Overdrive is really just used to improve fuel economy as it causes the engine rpms to lower, reducing the amount of fuel used. Turning off the OD won't mess up your tranny. Just don't do it too much, or you're gonna go broke paying for gasoline