Comparing Dates within plugin using PHP If statement

$current_date is a timestamp and $compare_date is a formatted date, so you’re comparing different things.

As you’re already using DateTime and as of PHP 5.2.2 DateTime objects can be compared using comparison operators, you can compare instead $expired_date (DateTime object) with today’s date as a DateTime object:

$current_date = new DateTime();
$expired = new DateTime();
$expired->setTimestamp( get_the_time('U') );  // get UNIX timestamp of current post
$expired->add(new DateInterval('P2D'));  // add 2 days

$day_color="";
if( $current_date > $expired ) {
    $day_color="#BA0000";
} else {
    $day_color="#39AC00";
}

echo sprintf( '<span style="float: right; color: %s;">Due Date: %s</span>', $day_color, $expired->format('F j, Y g:i A') );