According to the PHP documentation for timezone_identifiers_list(), that function should be included in PHP 5.1.x. The note on DateTime installation mentions, however, that it was only experimental support, and had to be compiled specifically to support it.
The fix, then, is to recompile PHP 5.1.x with
or to upgrade to PHP 5.2 where it is enabled by default.
My particular problem surfaced with some Drupal code that required the function.