The only time that my barrels tend to heat up is only when at the range (especially on a hot day) and never when hunting. Also, the amount of shots that it takes to get a barrel hot depends on the cartridge and the thickness of the barrel. You can shoot a .22LR all day long and the barrel might warm up at best, or take 10 shots with a 7mm Rem Mag and you can burn your finger when you touch the barrel.
Getting back to your specific question, whether permanent damage can occur when you barrel gets hot, I have no scientific data to give you. I can say this - whenever my barrels get hot, their accuracy gets worse. In fact, the hotter the barrel, the worse the accuracy. I own many guns from different makes, and this applies to all of them. But does it cause permanent damage? I really can't tell you, but I would imagine that when a barrel gets really hot, it does so not only from the heat of the powder, but also from the friction from the bullet going down the barrel.
I also tend to think of hot barrels like hot engines. With engines, when they get really hot, then it can cause serious damage. The hotter the engine, the more wear and tear, and the faster it can be permanently damaged. A barrel is also a metal. The hotter it gets from the powder and the friction from the bullet, then I think the faster it can be permanently damaged as well.
Zachary