Where does the air for the woodburner come from? Is there a dedicated air vent in the same room as the woodburner and is it open?
If not, then the woodburner has to draw air from the house which means that the air drawn from the house has to be replaced from somewhere and will leak in through any opening there is. If the house is old and leaky, you won't notice it. If the house is well sealed, then it can be a problem.
If you turn on an extractor fan then the situation is reversed and it is quite possible that the extractor fan will draw air in through the woodburner flue. If the woodburner flue is full of smoke then the smoke (and any CO) is going to get pulled back into the house.