Imagine the loads placed on the tire while making a sharp fast turn...it would try to roll the tire off the rim, right? Well *don't* try and take it off that way! Not even by standing on a 2x4 piece of wood to try and force it off. The tires have been designed to stay on the in the situations above.
Now imagine pushing the most inner part of the tire towards the middle of the rim. Use a non-marring tool (probably not a screw driver) and put it between the tire and the rim, then apply force so that it moves the tire inward. It won't take much effort at all.
In the end, this is a matter of applying little force in the correct spot, not getting a bigger hammer to force it off.
Best,
-Jim