It's really not recommended to use php code in Smarty. In fact it's now deprecated and you should avoid such solution as much as possible because it often doesn't have sense.
However if you really want to use PHP in your Smarty files for some reason you need to use SmartyBC (Smarty Backward Compatibility) class instead of Smarty class.
So for example instead of:
require_once(_PS_SMARTY_DIR_.'Smarty.class.php');
$smarty = new Smarty();
you should use:
require_once('SmartyBC.class.php');
$smarty = new SmartyBC();
Then you will be able to use PHP in your Smarty template files
EDIT
If you have just problem with including it's probably your directories problem (however you haven't showed any errors).
I assume you have your files inside your templates directory and you set it properly using:
$smarty->setTemplateDir('templates');
if you simple display index.tpl file in Smarty and you have this PHP file in the same directory (in template
directory) you can include it without a path.
However if you include in this index.tpl
file another tpl file, when you want to include php file you need to pass the full path to this PHP file, for example:
{include_php 'templates/file.php''}