My problem is that I don't know any way to prevent humans from mining iron other than making it extraordinarily scarce, and I don't know if this will have spiraling natural consequences.
Humans require iron in their bodies - think blood. We don't function without it and inadequate supplies are medically dangerous, so in practical terms humans could not live on such a world. A species with some different biochemistry might.
Without iron being relatively common in surface layers humans can't function here and to have it reasonably common in the ground requires that it will appear in large-scale deposits at some point that are at least possible to mine.
So off the bat, no iron or exceptionally rare iron and humans is a non-starter.
You'll need to move to humanoids with a different biochemistry.
First, would a planet with so little iron that it couldn't be mined develop differently, geologically speaking? (No nickel of course, for a start. But I'm fine with that.)
Short answer yes, long answer, probably unlikely.
Yes, most things are possible if they're exceptional cases on the edge of the bell curve.
Unlikely because planets like Earth are made of material from nebular clouds (we think) which cannot avoid being Iron-rich (or more precisely, cannot avoid having expected relative abundances of elements due to the way they would be created). Once iron is in there (and nickel is also going to be), you will get planets with Iron and it's likely to form in the surface as a result of geological activity.
Second, could I make iron that scarce without radically changing the terrestrial biology, forcing my creatures to use a different less efficient mineral as an oxygenator?
Not in my view.
But as I'm also suggesting you can't avoid Iron anyway, there's really no need.
I would like to build an Earth-like world in which humans don't have access to iron, except for what can be scavenged from meteor impacts, or at least not before industrialization. This would make steel a sacred, "magic" metal taken from the sky, and force them to rely on bronze for daily use.
Although Iron, Copper, and Steel have been known for thousands of years, copper and iron alloys were the primary metals in normal use for most of that time.
Steel, although known of, was something like a rare and very expensive metal reserved for only the most important things: military weapons, vital tools. It would be like perhaps Titanium is today.
The problem you have is that humans are curious. If they find any substance they will immediately start trying to break it apart, mix it with other stuff and who knows what. They'll find Steel by accident or design and after that, because they are not great at keeping secrets, this will spread. Steel is going to appear.
In pre-industrial times to make steel (specifically) such a special thing for people you simply need to make it hard to get good iron easily. That could easily mean that the easiest deposits of quality iron were e.g. from a meteor and simply not easily mined.
Or you could simply have the natives incorrectly associate the iron deposit they work with a chance meteor or comet that "led them to it". Nothing like a good random astronomical event to play with the human mind's tendency to find connections when there really are not any.
You could make the discovery of the process of making steel an inspiration from the Gods. Perhaps the process itself was discovered by or controlled by a powerful religious organization - that would allow them to control it, limit it, and of course to label it "sacred". Maybe the dominant religion bans it and it becomes a cult secret. There are many ways to work this.
Even greed: I discover how to make steel. Do I (a) rush out and tell everyone or (b) keep it a family/clan/religious secret and try and make a lot of money and power from this knowledge? And if history tells us anything it's that (b) is the chosen option for humans faced with this "dilemma".
So I would suggest looking at social engineering and psychology to make steel this big deal in your world, rather than making a lot of implausible changes to force Iron to be so rare. Politics, religion, greed - these are much more powerful ways to alter the world than trying to rewrite physics and biology.