1.The thermocouple in milivolt units might not be sensing a flame and the flame might be getting blown out due to a faulty damper or dirt near or in the pilot light tub. I've seen old systems with a pile of rust sitting on top of the burners!
2. Your furnace might be getting older, so periodic maintenance keeps you from getting a hefty bill when it breaks down at supper time in the middle of a snowstorm.
3. There are lots of problems with condensation not making it to the pump or to the drain period due to backed up condensate lines. Condensation in the wrong places can rust your furnace out, destroy electrical components, etc.
4. The blower motor and squirrel cage can get gummed up due to the renter not changing the filter or not even using one.
5 Check for faulty gas valves and plumbing that is outdated. You can actually get blockage in the gas line if it's an old school design that will allow solids to interfere with the flow of gas to your unit.