Problems upgrading to PHP 5.5
What you do when you remove a PHP extension used from a module you installed is uninstalling the module and then removing the PHP extension.
In this case, you also need to do something more: Since you are using the APC module as alternative Drupal cache, you need to remove the lines added to the settings.php files. Probably you will find the following lines.
$conf['cache_backends'][] = 'sites/all/modules/apc/drupal_apc_cache.inc';
$conf['cache_class_cache'] = 'DrupalAPCCache';
$conf['cache_class_cache_bootstrap'] = 'DrupalAPCCache';

Alternatively, you could find the following ones.
$conf['cache_backends'][] = 'sites/all/modules/apc/drupal_apc_cache.inc';
$conf['cache_default_class'] = 'DrupalAPCCache'

In any case, you need to remove the $conf['cache_backends'][] = 'sites/all/modules/apc/drupal_apc_cache.inc'; line, and any line in settings.php that references the DrupalAPCCache class.
References

The README file for the APC module

Side note
This is one of few cases where disabling a module, or uninstalling it without deleting its files, doesn't stop Drupal from using it. In fact, Drupal uses the following code.
  // Allow specifying special cache handlers in settings.php, like
  // using memcached or files for storing cache information.
  require_once DRUPAL_ROOT . '/includes/cache.inc';
  foreach (variable_get('cache_backends', array()) as $include) {
    require_once DRUPAL_ROOT . '/' . $include;
  }

It doesn't check the files are for a not enabled or not installed module. As long as it finds the files, it uses them. If then the code is using a non existing function (as in your case), it will raise a runtime error.