Our gas heater is 7 years old, and has worked fine in the two years we've rented here. This winter though, we can't get it to work consistently. When on "Auto", it doesn't turn on at all, no matter what temperature we select, and when we turn the fan to "On", it simply blows cold air. It occasionally decides it feels like working, but not consistently. Our outdoor temps have been varying from 32-65 degrees, so it's not as if we're in Siberia or anything, but I'm cold!

Our hot water tank is also gas, and we've had hot water, just no heated air.

Does this sound like something simple we need to do, such as look for some sort of filter to change or a switch to trip, or do we need to hound our landlord for a service call?

Sounds like the burners are not coming on at all. The blower is controlled by the furnace, it has to be above a certain temp in the heat exchanger or a certain amount of time after the burners light (about 90 sec.) before the blower will come on. You may have a bad thermostat (heat anticipator wire may be burnt) Take your thermostat apart and jumper from R to W if it works it's a bad thermostat.