I only occasionally remove a pinned barrel. As far as I am concerned, it is not necessary for routine cleaning. When the need or desire does arise, there is a distinct potential for damage, which can be minimized with careful procedures. Two gunsmiths that I respect gave me the following advice which has worked for me:
1) remove lock & tang screw using properly fitting screwdrivers. If improperly fitted screwdrivers are used, the screws will become "buggered up" and there is also a higher risk of the screwdriver blade slipping & marring the finish. Buy a set of gunsmith screwdrivers or grind regular screwdrivers to fit each size screw.
2) use a properly sized brass drift punch to gently push the pins out of the barrel lugs. the pins should have rounded ends to avoid catching and chipping out any stock wood and the pins should be a bit shorter than the length of the hole in the stock where they fit (the pins may well be of different lengths & if so, should go back in the same hole). By sitting slightly below the surface, the pins will not catch anything like a cleaning rag or hand that rubs over them and, the bit of hole in the stock wood past the pin acts as a guide for placing the drift punch. Trying to center a punch on a flush set pin is almost a guarantee of wallowing out the hole. My habit is to always push out from the left and in from the right.
3) with the gun upside down on a padded surface, lift the butt a bit & with the muzzle on the padded surface, gently "bump" to encourage the barrel to drop out of the stock breech end first.
In examining guns, you will often see where the stock is chipped out just behind the tang. This can be caused by heavy loads in a gun with a very tightly inlet tang or by removing the barrel by lifting the muzzle end, thereby making the tang a lever against the wood behind it. Be very careful with the stock once the barrel is out - they can be very thin and fragile. Swinging the empty stock into a ceiling or a light fixture could be a disaster.
Others will likely have their own tips - if un-necessary removals are avoided & necessary removals are done with care, you should have no problems.