Travis, dents need to be evaluated on an individual basis. But there will always be rational judgement involved, and no hard and fast rules. This is complicated by the fact that there are many different types of steel (or iron and steel in the case of Damascus and Twist) used to make barrels.
Barrels are thicker near the breech, and therefore it is easier to dent them further out where the walls become thinner. Lighter guns usually have thinner barrels, and really thin barrels can be much more easily dented in a fall or when struck by a car door, etc. But that is not to say they absolutely can't be dented back where pressures are higher. Raising dents where the wall thickness is greater is more difficult and takes more time. Dents always displace metal. Putting that metal back into place will naturally create some work hardening, but that is seldom a problem. Dents which can be seen inside the bore constitute an obstruction, and should always be removed before using the gun. Deeper dents are more likely to cause a crack than shallow dents. Dents near the ribs may crack or loosen the rib solder joints. That isn't necessarily dangerous, but the additional expense of rib repair may make it uneconomical. Dents near the ribs are naturally harder to remove because it isn't as easy to tap all the way around them with a dent removal hammer.
Some sharp dents are the result of the barrels hitting a sharp corner or rock, and the crease cannot be completely removed by simple dent raising techniques. When people draw file or strike the barrels on the outside, and perhaps hone them on the inside, to remove all evidence of such sharp dents, the barrel walls are naturally made thinner. If this is done too far back, or where your hands wrap around the barrels while shooting, it can create a dangerous situation.
That one is going to leave a mark which would have to be draw filed off and refinished in order to remove evidence of the damage. Opinions vary on just how thin is safe at various points along the barrels. There are numerous threads in the archives here with opinions of what is safe wall thickness.
