I'm a designer who's an expert in PDF's (stop giggling!) and hacking apart PDF's and similar types of documents is a small hobby for me. Granted, this isn't hacking in any real sense, but I certainly piqued my interest for non-political reasons.
Luckily for you, I've done a lot of the footwork for you and have posted my results below.
I just downloaded the form here: http://www.whitehouse.gov/sites/default/files/rss_viewer/birth-certificate-long-form.pdf and popped it open in Illustrator CS5. The article is correct in that it is not one flat scan, it is a composition. It's composed of 10 sub-layers most of which contain very small portions of scanned text.
More than likely, there were multiple scans made of the same document, and they were collated into this image or Optical Character Recognition was used for the scan. Editing isn't particularly surprising if certain sections needed to be touched up, but it is unusual.
It's not an issue really but, I don't understand why they didn't just flatten everything before converting it to final PDF.
For those interested, I made jpgs of the exploded document so that you can see exactly what is removed when the layers are sequentially turned off. Check it out here: http://imgur.com/a/Wy6li.
Maybe it's the equivalent of a clerical error, or maybe it's a political move to keep the birther's frothing at the mouth. Or maybe it's nothing at all. So it goes.