Over the years, I've bought and sold literally hundreds of tractors. Some with AG tires (most with AG tires in fact) some with bar type, and a few with turf tires.
My opinion....? There is no "hard and fast" generalization of which tractor/tire combo/application does the best/worst in all given areas.
Explanation....
Some BRANDS of tractors have better DESIGN MFWD systems. With-in the BRAND, certain MODELS are better than others. With-in the MODEL, certain makes of tires will do better/worse.
My experience with MFWD isn't limited to, but is based mostly on my use of a BIG John Deere (4955, 205 HP, MFWD, 21,000lb AG tractor) It has what Deere marketed at the time, a "positive caster" MFWD system. It leaves very little track (with front wheels), even with it's extreme weight. (18.4R X30 fronts)
In general, AG tires do more "damage" as they are more aggressive tread. They are INTENDED to dig more. Radials, and certain brands do LESS "damage" with equal (or in some cases, greater) traction.
Industrial (bar-type) tread does only slightly less "damage"than AG tires in most cases. Their purpose is to provide good traction, but with some slight sacrifice of mud/loose dirt traction, in favor of better wear on hard surfaces, or when extreme abuse is expected. (i.e., with a loader, causing wheel spin....)
The ultimate "non-damaging" tire would of course be, the TURF TIRE.... Talk about sacrificing traction for lessening "damage" to turf..... They are all about a light footprint.
Back to the issue of MFWD, tire damage, and there comparison to 2WD....
I'd say it depends on what the conditions dictate. A MFWD, in 2WD, with AG tires will leave slightly more "marking" than a typical 2WD in MOST cases. BUT..... When you're in a situation where the 4WD is NEEDED, that same MFWD, and it's better traction control might actually LESSEN the "damage" done by the front wheels.
One other "factor" to include is the HUMAN FACTOR.... I use mostly 2WD tractors in hay work. I can turn a tractor with-out scarring the hayfield 99% of the time. My 26 year old (hot rod) son plows up sod on about every turn WITH THE SAME TRACTORS.... Go figure....
Finally, soil conditions will have a significant effect on the story too. Wet, slick, or hilly ground won't react the same as a dry, flat piece of turf.
OK... Did I successfully dodge giving an answer, all the while talking in circles? I'm ready to run for public office if the answer is yes....