Shooting steel shot through a full choke MIGHT cause damage under these circumstances:
An older shotgun with a lower quality steel might get its physical choke properties changed from repeated pounding.
Ditto for a shotgun with very thin walls in the choke area.
Using early steel shot wads, which had thinner petals.
Or a combination of any or all of the above.
In a modern steel shot barrel, made from quality steel, with sufficient wall thickness, using modern thick petal wads, and limiting the larger shot sizes, the only damage that may occur will be cosmetic marks on the bore. The smaller the steel shot size, the less issue it will have. Note Remington's recommendation on not using steel larger than 2 through a fixed full choke.
Also note that this does not apply to their full Rem Choke that is marked specifically for steel. This is the one that has the golden tint to it, because it is made from special heat treated alloy. It works quite well with larger steel shot. If someone with an 870 was worried about their fixed full choke, the solution is to get a new or used 870 Rem Choke barrel and use the full Rem Choke rated for steel shot. Problem solved.
Another issue is blown patterns. Steel shot, because it won't compress, acts like it is choked one to two choke steps. Meaning steel shot in a modified will act like lead through a full or extra full. Steel shot in a full will act like an extra full or super full (ie "turkey chokes"). Sometimes this makes a nice pattern, and sometimes the pattern is blown. You have to pattern your particular shell and choke to know, and you cannot substitute another shell without patterning again.