Some of the finest bullets that are made have a full metal jacket (FMJ - which is actually a copper jacket, maybe you equate "metal with steel?). The fact that copper covers the tip of the bullet has absolutely zero effect on the rifling that it never touches. Thus it will not harm even the finest hunting rifle. As a matter of
fact, some of the finest hunting rifles fire a completely solid copper projectile.
The difference in FMJ bullet construction will have an impact on terminal ballistics compared to a PMJ (softpoint, ballistic tip, HP, etc.) - meaning it will most likely not expand as quickly or as much and probably have more penetration in game.
Differences in jacket thickness do vary but that cannot be determined by the designation of FMJ alone. Even the thickest or hardest will not substantially wear your bore more than one just a bit softer as Bart has tried to explain. I don't personally know Bart, but I have read a lot of what he writes .... He knows what he speaks of. Also, the construction of the bullet will make no difference to a fine rifle regardless of the action; whether bolt action, semi-auto, pump, break open or other.
There is already a lot of mis information here in this post, but we don't need to recommend the idea that a FMJ bullet is not suitable for for fine fire arms based off of what seems to be a poor interpretation of information.