Replace your filters if they're paper, or wash if they're electrostatic. Every six months. That's the thing that most commonly causes furnace failures. Ask this landlord how he knows that.
When the filter clogs, they will overheat and either melt the control board ($600) or you'll toast the blower ($200). Some furnace systems have multiple filters, so check both on the intake vent on the wall and inside the furnace unit itself where the air return enters.
Gas furnaces are pretty mechanically reliable devices. They're just jets and a sparker and a big fan. The sparkers and jets don't seem to break, but I have seen malfunctions related to 20 year old thermostat wiring jiggling loose. You should be able to visually confirm that all of the gas jets are lit, just by looking in there while it's running.
If you have one of the newer high-efficiency units, then you probably have a plastic (rather than metal) vent pipe coming out of the top, and it produces condensation. That moisture has to be removed, and depending on the geometry of your unit you might have a gravity drain line that drips it out under your house, or a pump system that uses an electric pump to remove it. Those electric pumps are intermittent and will typically fail years and years before the gas furnace does, which can cause all kinds of problems. I'd find your condensate line and figure out where it goes, because that's probably the second most common failure point after the air intake clogging.
If you have a heat pump instead of a gas furnace, then you have ten other failure points to worry about. I'd call a professional.