Why does my dryer NOT stop heating when cycle ends?

I have to disagree with @hay yew on the heating element being the problem here; generally you get NO heat when the element has gone out, not heat that won't go off. This is an electric dryer rather than a gas model, right?

I'm not an appliance repairman, but from my experience fixing dryers, my first guess would be the timer mechanism; there are spring contacts that open and close at specific times to start and stop the heating element and if it's gotten old/melted/corroded it the contacts may be stuck together and not opening when they're supposed to.

It is possible to open the timer up and examine it, but you do have to be careful as there are a ton of little parts that will fall and pop out. Last time I did it took me forever to figure out the proper way to replace all the little wheels and springs!

The nice thing about those older models like your is that they're pretty simple devices and a little Googling will usually give you the answer to whatever problem you're having with it.

I'd suggest checking the timer as your first step, though. That's what controls when everything happens, so that would be my first stop in diagnosing the problem. If it's just stuck contacts, it may be possible to pry them apart and take a file to smooth them back down.