Barrel life is so dependant on how the barrel is treated its hard to predict but I would on average say you could get 800 round easily for big game or large varmint accuracy in a 6mm-06. The key is do not let the barrel overheat.
Some say a three shot groups is useless, those will use up more barrel life developing loads then anything else. Three shots in this class of rifle is plenty for groups. Plus, at the third shot, the barrel is ready for a break as far as bore temps. If you put that fourth and fifth shot through the bore you go from hot on the third shot to doing serious damage on the 4th and 5th shots, pointless and wasteful.
Also, if your one of those that have a case of "tinkeritis", that meaning those that believe there is that magical mystical load around the corner, DO NOT GO WITH ANY OF THESE. Most of my wildcats are not barrel friendly and I strongly recommend to my customers to find a "good" load that will do what you want and stop load development and get off paper at 100 yards. More barrels have been wasted on paper at 100 yards then from any other use looking for that "Best" load.
In all honesty, is you put the bullet on point of aim, there is no real difference between a 3/4 moa and a 1/2 moa rifle for a big game rifle or large varmint rifle. Its far more critical to get your drop chart accurate and your rifle set up to match your drop chart then it is to take 1/4 moa off the groups of your rifles.
If your rifle puts your first shot within 1/2 moa of your point of aim at 1000 yards, you will kill any critter from the size of a yote and up at that range.
Anyway, my point is this, with an extreme performance round, the main thing is to get the load that will do what you want to do and then get out and do practical field practice shooting and get off paper. Your rifle will last MUCH longer and you will get much better with your rifle as well.
DO NOT overheat your barrel if at all possible. IF you can not hold onto your barrel bare handed indefinately without discomfort from the heat of your barrel, your barrel is to hot, don't do that to your rifle.
In some rare hunting situations this will happen but its really rare. The key is to not do it on the bench when you have total control to just stop shooting for a while.
As far as barrel length on these chamberings. Many say that you have to have longer barrels to get the advantage of the larger chamberings, that is not true.
I have built enough high intensity handguns to know that no matter the length of the barrel, the larger chambering will be the fastest chambering no matter the barrel length. For example, in a 15" XP-100 barrel length, a 7mm Dakota will always best a 7mm WSM in velocity and a 7mm WSM will always top a 284 Winchester in velocity.
In a 26" barrel length, a 6mm-06 will always top a 6mm Rem in velocity as will a 6mm Rem top a 243 Win in the same length barrel.
The only advantage is that the longer the barrel is, the more dramatic the velocity gain will be for the larger chamberings but its not needed or required.
Remember, the 6-06 will be loaded to around 54-56 grains of powder which is alot of powder in a 6mm bore but its not 100 grains of powder.
Most will way a 25-06 will do extremely well in a 24" barrel length, yes the 6mm-06 is more extreme but not dramatically so do not fall into the barrel length issue, to be honest, its not as important as many would think.
My personal 6-06 has a 27" barrel length. I worked up loads with RL-22 that got up to 3500 fps with no primer pocket loosening at all until three firings on the case. Obviosuly this is still a high pressure load so I dropped back down to 3400 fps. This is more velocity then most would believe a 27" barrel length could get with this class of chambering.
If the barrel was cut to 26", you would likely loose 35 fps at tops.
26" of barrel length is plenty, again, remember we are only burning 55 grs or so of medium burn rate powders so do not think it will not perform in this length barrel.
Hell, for the 6mm AI, the best powders will be pretty much the same identical powders as the 6mm-06 so whatever barrel length it would be limited to, so will the 6mm-06.
Again, my only concerns with the 6mm Rem or its AI version is not performance, or accuracy. Its simply that it requires fireforming and that for a hunting rifle, why not get as much performance as you can? You will need a long action anyway if you want to seat long range bullets out to where they should be to not take up so much case capacity so why not go with a long action, use a chambering that feeds as well or better and has plenty of capacity to do anything you want without running pressures to redline.
Just my thoughts. More has been made about barrel length dependance then really exists. Its relatively common thinking that you will loose 50 fps per inch of barrel length. I hear this all the time from new customers coming in. In reality, of all the chamberings I have tested in different barrel length, the only one that came even remotely close to this is my 257 Allen Magnum loaded with a 100 gr bullet weight and it would average 42 fps lose per inch of barrel from 30" down to 26 fps.
Even my huge 338 Allen Magnum only looses around 35 fps per inch from 30" to 26 fps, again this is on average.
In reality, conventional chamberings, even the extreme ones will only loose from 15 to 25 fps per inch of barrel length from 30 to 26" in barrel length.
Simply put, 26" of barrel length will perform great for you in any of these three chamberings.