One important thing that many fail to realize is the fact that, even the President of the United States has the same basic rights to privacy as does any member of this country. Granted, as he is a public figure, and in the public eye, that expectation is not a great as someone such as the people on this board, but he has privacy rights just the same.
To many, producing your birth certificate does not seem like that big a deal, and on the surface that belief would be correct. However, upon a more in depth examination, one realizes that something else would arise from it...setting a precedence! In short, if President Obama were to cave in this incidence, he would be opening the door to stranger and more intrusive requests for himself, and future Presidents. Once you open the door, even slightly, it's hard if not impossible to ever shut it again.
Finally, the fact still remains that President Obama is in a no-win situation regarding the whole birth certificate matter anyway, because those who claim they "don't believe" will never be convinced regardless of what President Obama were to produce. They would claim it was a forgery or something like that. Thus, nothing he could do will put the matter to rest.....in large part due to the fact that the whole driving force behind the "birther" movement isn't where President Obama was born to begin with! So what is the point again exactly?