Dryer shutoff repeatedly. Drum turns whenever door is open. Dryer will run for 3 to 4 minutes and shutdown

Sounds like you've got a messed up door switch. Maybe gummed up or something.
The dryer should stop when you open the door. And if this switch was to open at random invervals then it would cause the symptom you are describing.

I removed the back cover but didn't see any loose connections. I started the dryer and cycled through a load of clothes. Where is the door switch?

I am not sure, but sounds like a bad heating element, might be a thermostat also, very dangerous situation though, we are all human, you or someone may leave it unattended, please don't do that, even with a new dryer, they cause many house fires, if dryer is over 7 or 8 yrs old, consider a new one, life expectany is about 12 yrs, the door being open and the drum turning is another safety issue.

The door switch should be a little button somewhere on the front of the dryer, however it will be behind the door when you close it.

You could remove the door switch and ohm out (check resistance) on all the different contacts, "NO" contacts stand for "normally open" this means that when the switch is at rest, or nothing is pressing against it, these contacts are open. All dryer functions should go throgh this contact, in fact, the drum tunring, the heating element, the whole operation of the dryer hinges on this switch, of course it gets more complicated when you dig deep in the circuit, but you need not concern yourself with that now.

The other contacts should read "NC" this stands for "normally closed". When the switch is in rest position, (the door is open) these contacts close. Normally the only thing that will energize through this set of contacts is maybe an optional interior light.

Finally, you will see a common side, this is the side that voltage enters from. Think of the switch as a Y intersection on three streets, you are driving up on the "tail" of the Y, when you get to the intersection, you choose whether you want to go right or left. The switch works on the same principle.

Of course if any of this is a bit hard to understand, it might be best to let a pro handle it, or as Richard said, if the dryer has some years on it, you might want to consider replacement. Dryers are not something to fear, but they have to be respected, no safety should be neglected. To this day, I won't leave the house when a dryer is running, I won't sleep with one running.

Also, bear in mind it might not be the door switch at all, it would just be my first suspect in your situation.