My dryer just recently crapped on me. I turned it on before I went to bed and when I woke up it was still running. I had it set for 60 minutes, but apparently it did not shut off. Also it is not heating up anymore. Does anyone have any idea what the issue could be?

I had a similar problem with mine. The ignitor went bad on my gas dryer. Since the dryer wouldn't heat up, the sensor that detects the moisture content of the dryer wasn't satisfied, so the dryer kept tumbling.
Check your local phone book for 'appliance parts' stores. A good store will test parts and help with the diagnosis. Fixing a dryer isn't too difficult and usually worth it if it's in decent shape.

sboating. Is you dryer gas or electric? What model is it. Did the timer advance? You need to give as much information as possible