PHP version changes and Drupal versions issues
It's impossible to say whether your specific setup, with your specific site, will or will not have problems after a PHP upgrade. You'll need to set up a test environment that mirrors your live server as much as possible - and test it. It's unlikely, but you may have contrib modules installed that rely on PHP 5.2, so you need to check your specific installation.

From my own experience, before I upgraded to 5.4 some months back I was running many, many Drupal sites on 5.3 (though not versions as old as 5.3.3, that's pretty old these days). In over a year on dev/test/live I've never seen a single Drupal problem that was caused by the PHP version.

On another note, you really need to update your version of Drupal. We're currently on 7.22 and there have been a lot of security/bug fixes since the release you have installed.

You might also want to have a look at the PHP5.3 compatibility group, there's more detailed information there.