It's not that the Birth Certificates expire after 10 years, it's that the State Department want's to see a copy that is less than 10 years old.
10 years ago, our world wasn't as automated as it is today and the methods of certifying copies of birth certificates have varied. For example, the "original" copy that my parents recieved when I was born does not have a raised seal. It was "certified" by virtue of a colored stamp, the idea being if someone tried to make changes and cover up the changes by running it through a photo-copier, you'd know it's not original because the certification stamp would no longer be colorized. Obviously such a document would not be considered as secure today because of the advent of high quality color copiers and printers.
So (and I'm guessing here) because things have becomed a little more standardized in the world today, as a method of cutting down on phony "certified" birth certificates, the State Department wants copies that are "younger" than 10 years.