No burning firewood, no smoke of any kind, and no Febreze. Keep things well ventilated, watch out for drafts though. You might want to invest in a HEPA air cleaner actually. Good luck!
Birds are particularly susceptable to airborne toxins. I would imagine that fireplaces are safe- but I certainly wouldnt keep the cage so close so that it is affected by the smoke from the fire.
I think it is a much safer option to house the birds in a different room if you have an open fireplace.
It can if they are in it all the time. We burn wood but the birds are up stairs and in a bird room so they don't get the smoke, I use an oil space heater that run by electricity and is very safe. I do have the birds out in the same room as the stove but not for a long time.