I have very little experience with shotguns, but from the rifle side of things it very much can be done provided your smokeless ammo is correctly loaded. In rifles, most factory smokeless loads have a peak chamber pressure that greatly exceeds that of the same case loaded with black powder. Ergo, if you put the factory smokeless ammo into your BP-era gun, you are subjecting it to signficantly more breech thrust and barrel hoop stress than it was originally designed for. If you load your own (or buy from a company that specifically loads nitro for black ammo) it is plenty safe.
I would wager that the same is true for shotguns - modern factory ammo is higher pressure than a BP load, so by shooting the gun with modern factory ammo you are subjecting it to stresses higher than it was designed and proofed for. Some companies load super low pressure shells that are safe in BP-proofed guns, assuming they're in otherwise good mechanical shape.
