Gas or electric furnace ? How old is the unit because in last 10 years these no longer use PSC motors but a full variable speed (ECM) or an X13 motor that soft starts then speeds up to save energy but that's a laugh cause what I used to get to replace these when out of warranty would buy a lot of electricity for a PSC. control board or bad motor, if the motor it will die. Check tstat wire terminals for tightness at unit & tstat
If it's in a closet over a basement or 1st floor may be a pan & float switch under platform. I went on a call once that I found the filter in a wall frame so dirty when blower came on the air was pulling the switch open.
Electric heat. Date on the unit is 2008 if I remember correctly, maybe 2007. No basement here, it’s in a closet. Never have a problem with it turning on with the heater, only air conditioning.
That one may still have the time delay fan switch instead of relay. I just put a standard dbl pole/single throw relay as the reapair part.im leaning towards a loose terminal on the low voltage & fan is normally the green wire.