Without more information, I assume that the gas furnace is in the basement, the wood-burning fireplace on the first floor, on an exterior wall, that the home is rather new and air-tight, and that the fireplace chimney, when operating, causes underpressure in the home that is being equalized by air inflow through the furnace chimney flue.
In order to stop the air inflow, you can try three things:
- create a barrier between the air space that the fireplace draws from and the air space that the furnace sits in, for instance by putting weather stripping on your basement door to reduce the amount of air being sucked up from the basement. Home Depot sells an aluminum treshold with a rubber seal that will close off the bottom of the door. This could reduce the air inflow enough so that the pilot stays on.
- open a basement window, or install a permanent air duct in the basement with a damper that you would have to open before lighting a fire
- buy an automatic damper for the furnace vent pipe (something similar to this) and have it installed by a licensed electrician. When the furnace is off, the damper shuts off the airflow, and when it is burning, the pilot light can’t get blown out. This could be your best bet.
I hope this helps.