At normal cooking temperatures, Teflon is inert and stable. At temperatures above 570°F (300°C), the coating on nonstick cookware will start to break down, releasing toxic fumes into the air. These fumes are not healthy for people to breath, but they are lethal to birds even at very low concentrations. So if you or someone else accidentally burns a pan in your kitchen, your birds might all spontaneously die, even if they are in a different room or behind a closed door.
Self-cleaning ovens, certain kinds of heaters, and a few other products can have the same effect, but nonstick cookware is the most common source.
Safest practice is to not keep nonstick cookware in a bird-friendly home. Even if you are very careful, accidents happen. A common way this could happen is if a pan is forgotten on the stove. Obviously, that is a bad thing to do regardless, but made even worse if it is non-stick.