Login button not showing after upgrade to PHP 5.3
To diagnose the problem, make sure you display all errors (notices included) and that they reported somewhere (log, watchdog, ...), it should be a function incompatibility. Usually it is about parameters that are passed by reference for certain native functions or something like that (introduced in PHP5.3).

Then when you have identified the problem, open an issue on the module project (I bet it will be a contrib module or one of your custom module) so that the module maintainer can update it.

Drupal 6 is relatively old and versions prior to 6.14 are not working with PHP 5.3. Make sure you have the latest version, see also http://drupal.org/requirements/#php.