Primarily it sounds like a bad timer. You may have a secondary problem causing the heat not to work, but it'll be hard to tell until the timer is replaced.

I had this problem on my kenmore electric dryer. I can't remember if the timed settings did advance, but I do remember that in general, the timer wasn't advancing, AND the heat wouldn't come on. Both problems happened at the same time. I took the back off the dryer and found out that the heating element was dead and needed to be replaced, so I replaced it-- and the timer started working properly again. I didn't have to replace the timer at all. My guess is the heat somehow triggers the timer to advance on the non-timed settings, or the electricity that goes through the heating element also triggers the timer to advance. Long story short, replace the heating element first (if you know it needs to be replaced), then check to see if the timer is working.

When I pulled out my heating element, I could visually see the break in the heating wire. I could tell for certain that the continuity wasn't there. There was no need of a any kind of electrical instrument to check for it. :)

Good luck.