Start at the floor in front of the passenger's seat right-side (for LH drive cars). You will find some rubber tees with many vacuum lines. That is about halfway in the door/trunk lock system so can split to find the culprit. Split the tubes, plugging the side going left to see if that fixes your issues. With a hand-pump, you can insure each tube going right holds vacuum. My guess is the vacuum trunk lock is leaking, since a common problem. There is also a vac actuator for the fuel fill door lock (pull out trunk liner and feel up there). If a problem, I would just cap that off and live without that feature.

Your wording confused people. When you say that actuating the door lock made the engine shut off, you should have added "..., as it should have since the key was off".

Getting ready to dive into the vac/lock system on my '83 240D. This thread developed into an excellent guide and source of info! Specifically, great thanks and a big shout out to "Diesel911" for the troubleshooting link. That's great info there and a big help as it also links to door panel removal.
Personally, i think the biggest problem is not having an organized, methodical approach to repairing the system and this thread helps greatly to achieve a systematic step by step.
As mentioned by a couple members, you don't wanna stumble around in the dark, ripping everything apart, guessing if it could be this or that.
My system, for the most part, pretty much works "as advertised". However, ya gotta be quick. After shutdown, only have about 20 seconds, or so, to lock the doors. The vacuum will be lost after that. If locked immediately, everything locks and stays locked including the trunk. Recently noticed, and another "tip o the hat" to this thread, the fuel door pintle will extend upon locking, but quickly leaks down and retracts. I'm thinking this might be the crux of my problem and was gonna "start" there. As a member suggested, just plug this "leg" off and go without. Any thoughts? I have a "mighty vac". (bought about 40 yrs ago, one of the BEST tool purchases I EVER made!!)

I own a 1984 & 85 300D and have made junkyard picks. In my experience, the trunk lock actuator often fails to hold vacuum. I have maybe had 3 failed ones in-hand. I haven't seen a fuel door actuator fail. Rough statistics, but might save you time. The "leak-down" you experience could be due to anything in the "open" or "close" circuit, not just the fuel door actuator. I don't know why it changes position since I thought it is a push-pull actuator with no internal spring. Anyway, not needed unless really worried about diesel thieves, and smart ones would just cut the hose below to drain your fuel instead of fool with siphoning.

Just a thought, and keep in mind, I really have no idea how this system all works together. ....if the trunk actuator was leaking, would the trunk remain locked? Similarly, why do the doors stay "locked" even though I've, apparently, zero vacuum in the system?
When the doors are in the "locked" position, I can't unlock them, from inside, by pulling up on the lock/unlock stem. What force holds them down even though there is no vacuum present?

The fuel door actuator is unique because it has only 1 connection and it's to the locking circuit.

It also has a spring as a safety feature so that if you loose vacuum the fuel door unlocks itself and you're not stuck.

Other actuators don't do that. They stay either locked or unlocked whether you loose vacuum or not

Thanks for that clarification!! Makes sense about the fuel door. So, if I understand correctly, it's not vacuum that keeps the doors "locked" or "unlocked". It's simply the design of the locking mechanism and it's the vacuum that either locks or unlocks said mechanism? It would be great if there was a troubleshooting logic tree for this system (if there is, I sure can't locate it). However, considering the age of these cars and the fact that the door panels/seats/carpets aren't too difficult to remove, probably best to "expose" the entire system and see exactly what's goin on and where....

I wouldn't start by removing everything.

There are main junctions located in the passenger and drivers footwell if I remember correctly.

Start testing each line with a mightyvac and see which one holds and which one doesn't.

You will know where you have a leak and where you don't so that you don't make more work for yourself

Sounds like sage advice and a good place to begin. Thanks!