If the pans get overly hot, they are supposed to outgas something toxic to birds. That was the warning when I had indoor birds. It was heavily recommended that you not use teflon with cage birds in the house. however I never had a problem. I did read lots of testimonials saying they killed all their birds when they forgot the teflon on the stove.
I had a thread about it last year, and was told by many that the biggest danger was the pan getting too hot and outgassing fumes near where chicks were being brooded. But yes, if it starts losing it's coating, get rid of it. I have been relearning how to cook in plain old stainless again. Not worth the risks to use Teflon.