Steel vs. lead. Not likely to be a problem, unless you do something like plug a muzzle with mud and then try to shoot a slug through it. I've always been leery of shooting slugs through the various muzzle attachments that appear on some shotguns, like PolyChokes and Cutts Compensators, but that might be an over-reaction on my part. I've heard of improperly installed choke tubes or tubes with a dinged skirt being forcibly parted from the barrel and sent howling downrange by a slug, but have never seen it happen myself. Some companies warn against shooting slugs through their over-bored barrels - I'd follow their instructions on that one.
But for normal slugs of the proper gauge and hull length fired in a shotgun in good mechanical condition, no - damage to the barrel from shooting slugs is not going to happen. They were made for each other...