will shooting shot through a rifled barrel cause any sort of damage?
Most automatically says no, but they are probably thinking of standard lead shot, and the fact that there is probably not widespread experience doing so because the patterns are so horrible.
I would venture that the steel shot required for a significant portion of shot hunting would drastically shorten the life of a rifled barrel. A lot of small hard pieces of steel would be directly impacting a lot of rifling. It may or may not eventually lead to small chips in the rifling, but even if it did not it would certainly slowly remove the rifling.
Even with a shot cup some shot can be ahead of it directly impacting the rifling, and as the rifling bites into the shot cup it will touch steel pellets still in the shot cup grinding steel against steel.
Steel or other hard metal bullets in rifles and rifled handguns significantly reduce barrel life and destroy rifling. That is why they are typically only used for cores or have a significant layer of some softer metal or something like teflon that is at least the thickness of the rifling it is intended to encounter.
I do not think a shotcup will reliably perform that role with loose shot, nor be thick enough to prevent all contact of shot with rifling.
However since rifled barrels create doughnut shaped patterns that are horrible for any type of shot density in hunting, I imagine hunters loading rifled barrels with steel shot is a very rare thing.
Even an unchoked cylinder bore barrel is rare in the most common bird hunting (though fine in thick brush where the range would be close), and a rifled barrel is many times worse than that.
You won't hit what you are aiming at with a rifled barrel very much as the pattern density is minimal in the center and doughnuts, and the centrifugal force pulling the shot outwards will inhumanely wound things around it in a relatively uncontrolled manner.