A gas fire outputs carbon monoxide, if you believe there is any chance of
the chimney/flue being in anyway blocked or obstructed by a birds nest
have it throughly checked out before lighting your fire by a registered gas installer (rgii.ie)
Flue obstruction can easily cause flue spillage form the fire back into your living room and is a serious threat to life when they do. Do not take the risk.
As for the birds not minding the flue gas, if they were actually in the flue, maybe 30 mins to live before being overcome by CO