It usually goes away within an hour or so when I run my heat. Never bothers the cats, but one year it did set off the smoke detector. I later moved that particular detector further away from the duct.
It is probably a good idea to have the ducts cleaned professionally once every couple of years, and - of course - to clean and/or change all the filters in the system (depending on type) at least once a year. But a certain amount of dust is inevitably going to collect and produce a smell when you run a heating system for the first time - especially one that is rarely used. (I live in southern Florida and probably need to run the heat a grand total of four or five days a year.) It isn't so much that the dust is burning, in most cases, as that it is being warmed up and blown into the room - and the warm particles excite your smell receptors more than cold ones do, just as warm foods smell "more" than cold ones do.
Regards,
Joe