Say, if a character with Resist Fire 20 ran into a burning building to search for survivors or items. He would take no damage from the flames, but would he still suffer ill-effects from smoke inhalation, or simple lack of oxygen as it's consumed by the fire?
Yes, if the DM wants it to. No, if they don't.
Fire resistance protects against damage with the type 'Fire'.
There aren't specific rules for smoke inhalation, so the DM needs to decide how/if to model the problem.
You could just ignore it if your group is fine with that.
You could model it with fire damage, which I think would be a little odd, but do what you like. If you model it with fire damage, then yes the fire resistance resists it. It's fire.
There are rules for holding your breath, which is similar to smoke inhalation, due to the lack of oxygen. You could use these unaltered. This doesn't cause fire damage, so fire resistance won't resist it. It's not fire.
Smoke inhalation is worse than holding your breath, so you might consider increasing the DCs or the damage.
4e does not attempt to simulate reality.
Effects should have mechanics that relate to their role in the story, rather than what would be "realistic" (whatever that means in a setting with magic).
- If the fire is meant to be impassible, then it probably does a lot more than just 20 fire damage per round, as well as having additional negative effects not covered by resist fire to drive the point home.
- If the fire is meant to be a major obstacle but not insurmountable, then resist fire 20 would probably be enough to protect a character from all the fire damage but the smoke and other hazards (such as parts of the building collapsing) would still be a problem.
- If the fire is meant to be a minor obstacle then resist fire 20 is probably enough to ignore all of its effects entirely.
If you want the smoke to be a problem, then it would most likely be an attack against fortitude that inflicts one or more of (depending on how nasty you want it to be) blind, slow, daze, or ongoing poison damage. The building collapsing would probably be an attack against reflex or AC that does untyped damage. I believe the PHB has rules for using the Endurance skill to hold your breath if you want to add an additional problem in the form of lack of oxygen.