Rust and ash in my AC/furnace - should I be concerned?
I would talk to your landlord to ask them to inspect your furnace/AC for the year and let them make the determination.

When I do something like this with my landlord, I remind them that making sure things are right will save them from a catastrophe, possibly in the near future.  And yes, I do say it something similar to that.  I hold back only enough to let them feel as if the decision is theirs.  I also make valid statements about things should work, vs how it currently is working, even if I have to research that info first (like you are doing).

If your landlord is honest, and not a slumlord, you have a good chance to get at least some work done to it, if not replaced.  Most furnaces are designed to last around 20 years, so it's a good chance it's been replaced sometime in those 34 years since the building was built.  It may simply need a good cleaning and some basic maintenance.

Furnaces usually smell of burnt rust and dust when they first come on for the season.  If that odor continues past the first week of steady use, then it's not normal, in my experience.

Again, letting your landlord make the decision will make you sound like less of a jerk tenant (or worse) and more of a concerned person who knows what you/they're talking about.  Just because you got your information of the Interwebs doesn't have to mean anything to them.  ;-)

Simply making them away of a situation before it becomes a problem will, eventually and hopefully, will get you on their good side.  I find that useful, myself.