I recently came across this same issue after upgrading Magento to a newer version of 1.9.
It should be noted that you should never modify core files, and the password validation should not necessarily be removed.
In my personal case, I restored the original function to the core file, however, you could just as easily extend the Mage_Admin_Model_User class.
The version of the function I used can be found below.
/**
* Validate password against current user password
* Returns true or array of errors.
*
* @return mixed
*/
public function validateCurrentPassword($password)
{
$result = array();
if (!Zend_Validate::is($password, 'NotEmpty')) {
$result[] = $this->_getHelper('adminhtml')->__('Current password field cannot be empty.');
} elseif (is_null($this->getId()) || !$this->_getHelper('core')->validateHash($password, $this->getPassword())){
$result[] = $this->_getHelper('adminhtml')->__('Invalid current password.');
}
if (empty($result)) {
$result = true;
}
return $result;
}