Burning that kind of oil volume can generate white smoke. If the white stuff lingers for a while and smells like oil, it is oil. Oil burns blue when it is actually being burned pretty thoroughly. Lots of oil will quench the combustion and leave the engine as white, heated oil smoke/mist. You might want to take care of this quickly as your catalytic converter is not going to be happy with the quantity of oil you have put through it.
A dry and wet (squirt some engine oil in the cylinder before the test is done) compression check will give you a good indication of what kind of event you are dealing with - the dry test will tell you the relative condition of your internal engine parts, while the wet test will identify where the faults lie in the weak cylinders. If the wet results are the same as the dry ones, the problem is likely valves. If it gets better wet, you probably have a ring or cylinder wall problem. The head will be expensive but the cylinder and piston repair will be much more expensive. The compression check will be cheap.
Good luck, Jim
1986 Euro 190E 2.3-16 (291,000 miles),
1998 E300D TurboDiesel, 231,000 miles -purchased with 45,000,
1988 300E 5-speed 252,000 miles,
1983 240D 4-speed, purchased w/136,000, now with 222,000 miles.
2009 ML320CDI Bluetec, 89,000 miles
1971 220D (250,000 miles plus, sold to father-in-law),
1975 240D (245,000 miles - died of body rot),
1991 350SD (176,560 miles, weakest Benz I have owned),
1999 C230 Sport (45,400 miles),
1982 240D (321,000 miles, put to sleep)