In order to compare them this way you need to convert the field to a unix time value. The return value of the field needs to be in a format that will be recognized by strtotime()
see http://php.net/manual/en/function.strtotime.php
and http://php.net/manual/en/datetime.formats.php
$time = strtotime(get_field('date_field_name'));
$now = time();
if ($time < $now) {
// .....
}
Edit:
Also, new DateTime() creates a date/time object which cannot be compared to a string. It may be possible to use this object but it will take a lot more code to extract the string you want from it. I can’t help you with that because I personally use the simpler functions for dates and time rather than using this object.
