Trouble removing a(n old) 700x23c tire
How about grabbing the tyre where it's started to come off with some pliers. Then put a foot on the wheel, and pull hard.

Good luck! I know this can be really frustrating, because I had a tyre that was tightly stuck before. If it really won't come off, buy a new tyre that looks more forgiving, and chop the old one off with some wire cutters or a knife.

I decided to replace my too tight tyre even though I eventually did manage to get it off and back on because I considered that if I ever had a flat away from home and all my tools, there would be no way I could repair it, and I didn't want to risk being stranded a long way from home.