When I was a teenager, my family kept zebra finches and button quail in a flight cage in the dining room. The kitchen was the next room over with an open floor plan. One day, I left a pot of water on the stove to boil and forgot about it. The water boiled off, the pan over-heated, and the button quail were found dead later that same day. Thankfully, the finches all survived. Perhaps the quail were more sensitive or inhaled more of the fumes. I don't know.
It was a really horrible way to learn about the dangers of non-stick pans.
It is a serious health risk for birds and kills very fast. Be safe and don't risk it.
It is also worth mentioning that the same toxic fumes that can kill birds are known to be harmful to people also. We won't die from low levels, but it is not good for people to breath these fumes regularly or in large amounts, as they are linked with multiple health problems. Even if you do not have birds, you should use caution with old Teflon coated cookware and be sure to avoid overheating the pans and always have proper ventilation. Modern Teflon is PFOA-free, so it should be safe.
And, of course, never walk away from a boiling pot of water and forget about it, like I did. Even if the pan isn't Teflon coated, it could cause a house fire!