I read someone was able to "remove" the click sound by playing the same click sound phase shifted when taking a photo - so when the OS sound mixer adds the two waveforms together it results in silence. Clever idea - it is the same as noise canceling headphones, except it all happens in software.
As others have mentioned, if you put your iphone in silent mode there is no click sound except in Japan. I'm not sure if a) phones sold in Japan have disabled this feature or b) the phone knows when it is located in Japan and then disables silent mode.
I doubt that would work well in practice. The reason is that timing is extremely important for this to work, if you're even one sample off you'll here a faint sound, and more than a few then it's quite obvious. So if the sound is generated by a separate OS process, you'll never know exactly when the sound is generated and you will not even have control over exactly when your own sound is played back in relation to the other process, think sub ms accuracy.
As others have mentioned, if you put your iphone in silent mode there is no click sound except in Japan. I'm not sure if a) phones sold in Japan have disabled this feature or b) the phone knows when it is located in Japan and then disables silent mode.