Assuming you're speaking about a modern shotgun with removable chokes... No, the barrel isn't an issue. Chokes are though.
The chokes are an issue when shooting Steel or Heavier-than-Lead because the shot pellets are hard and can't deform like the softer lead shot can.
Most modern removeable choke shotguns can do steel with the stock chokes offered by the manufacturer. But the stock chokes don't do heavier-than-lead. (Would cost more to produce the stronger material chokes necessary for Heaver-than-Lead shot.
Your choke narrows and constricts the pathway thru which the shot pellets travel. Therefore you must make sure the choke says directly on it that it is ok to use for Steel and/or Heavier than lead (Tungsten/Bismuth). Since those metals for shot pellets will not deform if constricted too much, not only is there an upper limit of how much you can choke them down... but you also need to be using a choke constructed of hard-enough metallurgy to withstand being subjected to those shot metals which are much harder than lead.
I did a lot of reading on this recently as I hadn't touched a shotgun in 30yrs, and recently decided to get back into it. For my purposes, I was concerned with using one for Predators. For the heavier-than-lead offerings, you need to read the information provided by both the ammo manufacturer (because they may tell you that you shouldn't put on a choke that constricts more than <some measurement> ) and also make sure the particular choke you are buying/using states specifically that it is safe to use with steel or heavier-than-lead.
Apparently, when you start getting into the bigger shot sizes in Heavier-than-lead... such as the HEVI-SHOT Dead Coyote in BB or T, for some of them they say not to go more that Modified, that it will actually make the pattern worse if attempt you go tighter than a Modifed. And that also, exceeding their recommended amounts of constriction can cause a dangerous condition (from the pressures that occur when these shot of harder metals are forced to collide with each other as they pass thru an area of too-much constriction in the choke, giving them no wiggle-room or place to go away from each other, as you can imagine, with the lead, when push comes to shove, some of them can deform as needed, that won't happen with the Steel or Heavier than Lead).