The serial number ranges of the iPod shuffle with the affected headphones are xx909xxxxxx to xx952xxxxxx and xx001xxxxxx to xx004xxxxxx. A very small percentage of iPod shuffle owners have experienced the problems, according to Apple. The issue is apparently limited to headphones that were included with the 3rd gen iPod shuffle made between approximately February 2009 and February 2010. Apple also notes that any Apple earphones with built-in remote, like the ones that ship with the iPhone, are covered under this replacement program as well.
Full details on the replacement program, including how to locate your serial number and the steps to go through to obtain your new headphones, are on the Apple Headphones with Remote Replacement Program page.
Thanks to reader Michael B. for the tip.