I've checked all the main components except the motor for continuity and they all check out fine.
The AC breaker gets tripped and when flipped back on the dryer works. Dryer breaker has not tripped.
All parts are original except heating element, which is maybe 3-5 years old.
Dries fine when running and will finish the whole cycle. It will only run that one cylce, though, and as far as I know it trips the breaker when the load finishes.
Update 2016-02-12:
The breaker labeled dryer is definitely the breaker controlling power to the dryer. Turning it off stops power to the dryer. I'm not sure which part is connected to the AC breaker.
It's too cold right now to turn on the AC so I haven't tested whether it will go off when the AC breaker is off.
I've investigated more and found that the door switch is broken, so it is not always engaged. Could the switch being momentarily engaged and then disengaged cause the issue?
Best Answer
Sounds to me like someone crossed two wires coming from the AC and Dryer cables to the breakers. They are most likely both #10's and could easily be confused for each other.
Try shutting off the AC breaker and see if the Dryer runs with its breaker turned on. This sounds like what you said is happening in your question but just to double check try this test. If the Dryer doesn't run than than what happened is two wires are crossed to the two breakers. One is feeding the black on the AC and the Red on the dryer and the other is vice-versa.
Either way though, there is something wrong with the Dryer if it tripping a breaker. If you are not comfortable shutting down your main and working by flashlight in the panel with the cover off to straighten out the wiring then you should call an experienced electrician to check this out for you.
Also, you will probably have to call an appliance repairman unless you feel like getting into your dryer and working on that too.
Good luck with your project!