I am planning on boiling a mixture of water and bleach to lighten fabric. Will there be fumes that are harmful to my bird?
What type of bleach are you using,You need to take a look at the container is it hydrogen peroxide or chlorine.
Hydrogen peroxide do only release oxygen and is not toxic,Hydrogen peroxide breaks down and do only release water and oxygen when heated,so it is safe to use in the same room as you and the bird are.
Chlorine do release chlorine gas and is very toxic to you and the bird,When you heat chlorine a large amount will go into the air(chlorine gas was first used during the first world war to kill the enemy).
please take a look here https://en.wikipedia.org/wiki/Bleach to learn more about the different types of bleach.