Should I turn off heat in my home when I am finishing my basement?
My cousin is an HVAC contractor and always recommends that people NOT have their HVAC running while drywall is going up in order to prevent all the sanding dust from clogging up the furnace/duct work. He even offers to temporarily install an old system that he's got if it's necessary to provide heat to the structure.
It will take heat for interior coats (drywall mud, paint, caulk, etc) to dry/cure properly, so you'll need to provide some sort of heat. Also, if you're living in the house, you will need some sort of heat. This can be temporary electric or propane heat if necessary. Be exceedingly cautious if using any sort of temporary gas heat to ensure adequate ventilation so you don't kill yourself (some consider this worse than getting sick), and that you don't burn down the house (remember, the propane is burning to provide heat).
You may consider temporarily blocking off all return air ducts coming from the areas being refinished in order to help keep the dust from being sucked into your heating system. You may also want to invest in higher quality HEPA rated filters for your furnace, at least while you're doing the renovation work, and change them more frequently knowing that no matter how well you seal off the work area, some dust will leak through into the rest of the house.