If I recall correctly, Bon Ami is a powdered-cleaner that is an "all-purpose" cleaner that is advertised as being "non-toxic" and "safe for pets and people". It's very, very abrasive, and the idea is that you can use it to clean anything and everything and get anything and everything that is "stuck-on" and hard to clean-off with it, without the use of chemicals because instead of using chemicals to "dissolve" dirt/grime/grease/etc., it uses the really abrasive ingredients to scrub it off.
As far as using it on your bird's cage, perches, toys, dishes/bowls, play-stands, etc. it should be fine, as long as you rinse everything extremely well and get all of the cleaner off of the objects (as with anything you use to clean your pet's stuff).
I would think that the only real risk to birds from this stuff, as Scott already mentioned, would be the dry powder getting into the air and your bird breathing it in or getting it in their eyes. It's extremely abrasive stuff (hence the limestone, feldspar, soda ash, etc.), and if your bird was to inhale even a tiny little bit of this stuff (or you) I imagine that it could cause a great deal of irritation and possible breathing issues, and may end-up resulting in an Upper Respiratory Infection. So I'd make sure that you don't ever use it in the same room that your bird is in, and it's probably best to even use the stuff outside, in a garage, basement, etc., or a room of your house that the bird doesn't go into with the door shut. And I'd be careful yourself too, as if you were to breath this stuff into your respiratory tract through your mouth or nose, or even worse, if you were to get it in your eyes, my god would that be irritating and drive you crazy...It could very easily scratch the lens of your or your bird's eyes...But as long as you're safe about using it, you keep the dry powder totally away from your bird, and you rinse everything you use it on very well, it actually sounds like a pretty safe alternative to using chemical-cleaners, which you should never use...