You'll need to post the errors.
Few things have actually changed between 5.2 and 5.3 that I would expect to cause a break. There are a number of improvements in the OO implementation such as late static binding though, so that could be a potential issue (although chances are that would not be).
I'd be more prone to lean towards an incorrectly configured 5.2 or 5.3 being a cause over that of a change in the version. Zend does put great care into maximizing backwards compatibility. The last major one was caused by 5.3, but that is related to namespacing and class names. 5.2 doesn't support namespaces, so that won't be a BC issue.
As of PHP 5.5, the MySQL library has been officially deprecated. It is recommended to move to either MySQLi or PDO libraries for your mysql connectivity. See here for help choosing which interface you prefer: http://php.net/manual/en/mysqlinfo.api.choosing.php