The answer is "Why would you take the risk at all in the first place, knowing that fumes/scents/smoke/odors effect all birds in a very serious and negative way, and that you may very well be hurting/killing your bird?"
You don't know what's in the incense you're burning, well, not unless you dipped it yourself. Same with burning candles, oils, etc. You have absolutely no idea what you're putting into the air, and as already mentioned, you can't see what is still floating around in the air...regardless, it really doesn't matter.
There is absolutely no reason at all to take a risk to your bird's health that you are fully aware of. Period. Your bird has an extremely sensitive and very unique Respiratory System, and their Air-Sacs alone make it imperative that they are not breathing-in any particulates in the air...and you're putting particulates all through the air when you burn incense, whether it's "natural" or not...So just don't burn it in the same room that your bird is going to be in within hours of burning it. Unless you think that being able to burn incense in your bedroom is worth risking the life of your bird. Then that's another story entirely, and you need to do some serious thinking...