Geez, I didn't hear/see anyone provide the tricks you use. Marty mentioned using a couple of flat screw drivers. Well, been there done that got the T-shirt to prove it AND skinned knuckles to boot.
Alot depends on the tires themselves. If it's a 2-ply rating it will be more flexible. 4-ply can be pretty stiff.
Ya, if you're throwing away the old one you can just cut it off with a sawzall or hack saw, or similar. For me, most of the time I was removing the tire to clean up the wheels and paint them, and then reinstalling the same tire or some other used ones.
I don't recommend doing this in the winter or cold temps. And I don't like the idea of soaking the tire in hot water since you can never really dry the tire on the inside - I didn't want any water around the new paint.
I used a big ole flat screw driver with an 18" handle, and a flat spoon like tool similar to what you see at tire shops. I think I got removal time down to about 1-2 hours along with thousands of cuss words.
Re-installation on a nice painted up rim without damaging the paint probably took 1-2hr x 2 or 3. Leaving it sit a few times, switching to starting and doing the other one, etc. I did get it done, but it takes a lot more patience than Marty implies, and some touch-up painting.