The spare wheel that I have will not let go of its tire. It looks like the rubber has actually rusted to the metal wheel. Perhaps the steel bands inside? I don’t know, but I’ve tried regular methods of tire removal, and when that didn’t work I started cutting the tire. First, with a hacksaw, then a box cutter/utility knife. Hard work, but I got most of the tire cut away. A ‘ring’ of the tire is still affixed to each rim of the wheel.
The wheel, once I started cutting off the tire. 
I’m left with two rings, stuck against the outside and inside rims. 
It looks rusty.
While searching the internet for someone with a similar problem, I found a suggestion that the tire may be loosened by saturating the rim area with brake fluid. I tried that over several days. While it removed the paint from the wheel, it didn’t do a lot to free the tire.
Next step — to clean up the area where the tire and wheel meet. Maybe I can see better what is going on. I picked up a wire wheel attachment for my angle grinder.
The wire wheel removed some tire material, until I could see the metal bands/wires inside the rubber of the tire. Eroding the tire rubber smells bad.
Once I got to the metal strands, I used a pair of lineman’s pliers, and then a cut-off wheel to cut through some of the bands. Once I cut enough of them, the tire ‘ring’ came loose of the tire. I didn’t need to cut completely through the tire. It seems that the circumference of the tire had tightened-up over the years. Getting it loose enough allowed the tire to then be pried off the wheel.
Cutting through some of the tire strands. 
And it’s finally loose!