You want a bird cover that you can throw in the washer and dryer so any heavy duty natural fabric like cotton or canvas will work. What kind of birds to you have because almost all the bigger birds will chew a cover. You want to make sure that the stitching is tight and tiny so they don't get their talons caught in it.
Ginger
Found this online...
But in the meantime, did you know that many popular fabrics are made of acrylic fibers? One of the most popular is Sunbrella outdoor fabrics. Sunbrella fabrics have been certified by GreenGuard Children and Schools because the chemicals used in acrylic production are bound in the polymer – in other words, they do not evaporate. So Sunbrella fabrics do not contribute to poor air quality, (you won’t be breathing them in), but there is no guarantee that you won’t absorb them through your skin. And you would be supporting the production of more acrylic, the production of which is not a pretty thing.