Of course there would be mayor attrition.
"Living off the land" basically means that you send your forces to find food for themselves and their mounts at the expense of the combat movement. It slows down your units considerably (loss of movement speed), reducess their readiness (loss of ORG) and of course, make good part of the division not ready to fight (loss of STR), as it's not present where it should be. Additionally, in case of any mechanized equipment (which is present not just in mech/mot/armoured troops, but pretty much every WWII unit) rate of attrition is even higher. Of course over time those losses will be gone (as forager parties and broken down vehicles will catch up at some point), but over the long trip, those losses really mount up and at some point simply stop whole combat movement.
On the top of that, in real situation you would have to leave people behind to keep some basic order on the occupied provinces along the supply line. As the supply line gets longer, the more soldiers get involved in keeping it up, especially if we are talking about 30 km wide corridor running hundreds km deep into the enemy territory. It shouldn't (and in reality, it never would) require whole brigade of regular or partisan troops to cut the line off. On the unguarded supply line, bunch of marauders would do the thing just as well.