This AFM (Active Fuel Management) system is standard on all the GM V8 gas burners starting in 2007. Cylinders 1,4,6,7 will be turned off whenever the computer (which is so much smarter than us) decides to do so based on speed, engine load etc. The fuel, spark, and the oil supply to the hydraulic valve lifters will be turned off to the aforementioned cylinders. The special lifters on these cylinders are spring loaded in order to keep the valve train from falling/flying apart when the lifters lose their oil supply. These lifters are subject to failure due to their more complicated design and the intentional and frequent loss of hydraulic pressure during operation.
Also since the valves don't open on the 4 dead cylinders, every time one of those 4 dead pistons is on the down stroke it creates a vacuum in that cylinder. This vacuum demands to be filled somehow and this results in "swabbing", or sucking, engine oil into the cylinder past the rings. So if you have oil consumption isolated to cylinders 1,4,6,7, now you probably know why.
The system is robust enough to usually make it past the warranty period. After that I would opt to program the AFM into oblivion so that it will not intentionally ever cut off the hydraulic supply to the lifters again. The constant on/off cycling of the hydraulic pressure is what leads to failure more than anything else IMO. There are options for this listed in my previous post. If you wait until the lifters etc fail, you're going to be going the far more expensive route of tearing into the engine to at least replace lifters. Unlike previous engine designs this will require cylinder head removal, and much greater expense.
If you opt to restore the failed AFM system to its previous glory, there are many extra parts to be replaced and the engine oil system will need a good cleaning. The slightest bit of trash/gunk in the system will lead to repeat failures due to the tiny passages in the oil control solenoids etc. It's highly recommended to use synthetic oil in these engines to prevent varnish and gunk buildup AMAP.
Some of the bigger speed shops, if local, can program your computer for you, or you can send it to one of the outfits that does this using an express shipping process.
If you opt for the complete delete you'll be replacing the spring loaded lifters with the same design that is used on the non-AFM cylinders. It's recommended to replace the cam as well, and the cover plate underneath the intake manifold which contains the control solenoids will be replaced with a blank plate. The outfit that sells the delete kit will reprogram your computer and return it to you while the vehicle is down.