What could be wrong with my dryer timer?
In general, if the timer isn't advancing, is making odd noises, not stopping, etc. the solution is to replace the timer.

I am not familiar with Tappan Appliances but the last time I replaced a timer on my dryer the timer fit several models from different makers and had the same connectors. Thats a clue that the timer is a very generic part.

You should search for general dryer timer diagnosis steps. For most dryers you can test the timer by checking for resistance or the flow of current between pins with the dryer disconnected from the mains. The steps will have you check for resistance with the timer in various positions. You are checking to see if the timer is turning the right circuits on and off when it is supposed to. If the timer is OK but the circuit is not coming on then it is a problem with another component on that circuit . If you can find a guide for your brand of dryer that is best, however any guide should give you an idea of what the timer does. At each position it turns a circuit on or off to control the heater or motor of the dryer.

Some mechanical timers may just get dirty. I think this is a possibility in your case. Taking the timer out of the dryer and looking for burned contacts or mechanical jams might spot the problem. Since your timer is buzzing it could simply be stuck, the buzzing might be the timer motor trying to turn a jammed connection. 

Nearly all dryers are fixable. They are a very simple product with a few simple circuits. You do need to exercise some caution. The circuits are simple but they do control powerful heating elements and motors and use higher current and higher voltage.  

I hope this general answer helps you identify and solve your problem. Be confident that you can fix this!