Most people notice this on the first genuinely cold morning of the year. The heat comes on and the house smells like something is burning. Nearly always it is nothing. Occasionally it matters a great deal, so it is worth knowing which is which.
A furnace sits unused for months and collects dust on the heat exchanger. The first time it fires, that dust burns off. The smell is dry and slightly scorched, it fills the house for maybe half an hour, and then it goes and does not come back.
If that describes it, open a couple of windows and carry on. Changing the filter first thing in the season reduces it.
Sharp, acrid, plastic-like. That points at the blower motor, a component that has failed, or something that has fallen onto the heat exchanger. Switch the system off at the thermostat and at the breaker, and get it looked at before running it again. It will not fix itself and it is not one to wait out.
Usually mold or mildew somewhere in the ducts or around the coil, especially in the coastal parts of the county where humidity sits higher. It is unpleasant rather than dangerous, but it is being blown through the whole house, so if anyone indoors has allergies it is worth testing the air rather than guessing.
Points at a leak or a component running hotter than it should. Worth a look, not worth panicking about.
Rotten eggs, or sulphur. That is not dust and it is not your furnace needing a service. Natural gas has an odorant added to it precisely so that you notice it.
If you smell that, do not troubleshoot it. Do not switch anything on or off, do not go looking for the source, and do not call us first. Leave the house, then call the gas utility from outside.
You are probably skimming this article, the way anyone skims an article about smells. That is the paragraph to keep.