Hi Diesel Benz; I really don't want to be the one to pee on your parade. But Buddy, I think the answers to your problem should be pretty evident. I was hoping a couple of the big guns on this forum would come to your aid and say, Hey, hey, hey, you just got to do a little more tweaking here and there and that little old 240D will be as happy as a little old pig eating "grunt".
But life is life, and it ain't always easy. So to start lets look at this little paragraph.
"we rebuilt the engine ourselves so I haven't put much money into this car yet (that's how I like it).

But why did it start burning oil suddenly?"
That is why it is burning oil. You rebuilt it yourself and didn't put any money into it. What did you use for parts? Junkyard parts for an overhaul? Probably not. But here is what I suspect. A set of rings for all 5, any rod bearings at all, well maybe one or two looked bad enought to replace, borrow a ridge reamer and hack that ridge out of there, get the pistons (old) with new rings past what is left of the ridge and now use those two new gaskets, one for the top and one for the bottom, crank it together and you are very lucky it started up for you.
Then you mention two oil changes, one at 5000 miles and one at 10,000. Everyone on the forum is screaming, "change that oil and change it often" and they mean no more then 3000 miles. Don't tell me you had Mobil 1 in there. After doing a driveway overhaul you should have changed the oil at least at 500 miles and then again at 1000 or 1500.
But I suspect the damage was beginning as soon as you started it up and continues even now, at least when you run it. Take it apart and I think you will find broken rings, and broken cylinder ring "lands", that is the iron ridge between the rings. And probably scored walls and perhaps a broken piston or two.
But hey, don't give up on it. I haven't seen it and I don't know too much. If the car is as nice as you say, and if you don't want to spend the $2500 to $5000 for a total rebuild look around a bit and you just might find a good used one that could go a long ways. There are quite a few around.
But I think these engines are way past the point where anyone can remove the head and pan, slip on "some new rings" in the driveway or home shop. Our Grandpa's could do that with Henry's Model T and a few other vehicles of that era, but that will never work on modern engines, especially the diesel.
Again I hope that you can take some straight talk. I really like this forum for the information we can get. But we still have to work within our limits - tough limits of fit and finish, almost all tolerences at .0001 or 2 and some more stringent than that and cleanlyness that almost matches that of hospital surgery rooms. Thats tough to find on the driveway!

"If it was easy everybody would do it!"

Junqueyardjim