Could the radiator reservoir be overpressured and leaking because of a blown head gasket? How about the crankcase? Does the oil have a “milky” look to it? I seem to recall certain early Neons had a problem with head gaskets.
I had a riding mower that one day decided to start BURNING oil! I don’t mean a lot of smoke. I mean big FLAMES coming out from under the mower. This thing had been nothing but problems from day one so I decided not to spend a second of my time or cent of my money looking into the cause. I had to add like a quart of oil every 10-15 minutes. This was a “Fill up the oil and check the gas” type of engine. Eventually it got to where it no longer had any compression/power and I just started adding oil from the jugs of used car oil I had sitting around to be recycled. It was only burning the oil for a while, then one day it decided to also start leaking it. Things got real interesting when the entire engine became engulfed in fire about 1/2 mile from my place. I knew that if I stopped it where it was, the entire thing would burn up. I decided to take the risk and try to ride it back to my place with flames coming out from all around. Somehow it made it all the way back and some wiring was burned so it wouldn’t turn off. The thing finally backfired to a stop after I had the fire put out. Eventually it got to the point where you couldn’t keep oil in it and it locked up. I have some nice videos of the thing shooting flames at night.
I have also seen some neglected Geo Metro engines that became real oil burners. 1 quart per 100 miles isn’t uncommon on some of the bad ones where people just added oil and never changed it for like 100,000 miles! The sad part is these things will run forever if treated right but they are not at all forgiving to neglect.