There have been many problems about the installation of the php-zip module. Since Mid of 2006 the php-zip module does not require the zziplib anymore - it uses its own implementation (which is a clean approach in a double sense - there are no source code comments). So, the following might possibly be only relevant for older installations.
Chris Branch has been kind enough to jot down the points of a successful php-zip installation sending it to me in May 2006. I am quoting his text verbatim - again, I do not know whether it works or not as I am not using any PHP for real work.
[***] Note: That step is the critical step that's not obvious. Apparently, when you build PHP as a static library and include the "--with-zip" option, it creates a static library for PHP with an external dependency on zziplib.a. However, the Apache configure script and resulting Makefile doesn't take this into account, so Apache won't link unless you hand-edit the Apache Makefile. (Maybe there's a better place to make this change so that you don't have to re-fix Apache's Makefile each time you run Apache's ./configure. However, I didn't spend the time to investigate that).