PHP Sorting Array by Time & finding the closest matches to X numbers

I tried to build a solution based on the approach that you took and made it much simpler. There are a number of edge cases that should be considered, and I will explain those later, but I think that it will still achieve the basic goal of your app as is, but I did make some assumptions.

...
<?php

    $counter = 0;
    $daily_max = 3; //how many unique feeds to display per day
    $display_interval = floor(24 / $daily_max); //simple way to make even intervals
    $posting_time = array(); //to store the times to post

    // Create a list of time intervals largest to smallest ex. [16, 8, 0]
    foreach(range($daily_max-1, 0) as $i) {
        $posting_time[] = $display_interval * $i;
    }

?>
    <dl>
        <?php 
            if($maxitems == 0){ 
                echo '<dt>Feed not available.</dt>';

            }
            else{ 

                foreach ($rss_items as $item){
                    if(count ($posting_time) == 0){
                        break;
                    }  

                    //as soon as the first item is older than the most current post time, output it & count that time slot as being filled
                    $rss_item_hour = $item->get_date('G');

                    if($rss_item_hour < $posting_time[0]){ 
        ?>

                        <dt>
                            <a href="<?php echo $item->get_permalink(); ?>" 
                                title="<?php echo $item->get_date('j F Y @ G'); ?>">
                                <?php echo $item->get_title(); ?>
                            </a>
                        </dt>
                        <dd>
                            <?php echo $item->get_description(); ?>
                        </dd>

                        <p>
                            <?php echo $item->get_date('j F Y | G'); ?>
                        </p>

        <?php 
                            $counter++;
                            array_pop($posting_time);
                    } 
                    else{
                        // Debug message
                    }
                }
            }

        ?>
    </dl>
...

Ok, so since I don't have access to your fetch_feed data, this is untested, but I am happy to update if there are bugs.

What this will do is pick posts that are roughly broken up by the interval that you specify, but it does not do any checking to determine how close they are to those boundaries. For example, if the latest post is at 16:01, it will be skipped in favor of the first post that comes before 16:00, which may be at say, 9:00. Then it will look for the first post that is before 8:00, which may be at 7:59, so then you will have two posts that are really close in time. Or if there are no posts between 16:00 and 8:00, the first post displayed may be at 7:30, and then the very next post, maybe at 7:28 will also get displayed (since it is now the first post available before 8).

My assumption was that you are less concerned about the exact spacing and are more interested in 'thinning' out the volume of posts a little, which this should achieve and is hopefully suitable for your application.

As I said, I am happy to help you refine it if you have something specific in mind.

Considering the time as "seconds of the day" (0 - 86400) the following lines would serve your needs (simplified example):

<?php

$postTimes = array(1,600,953,1900,23500,27600,56000,72000);

echo "Closest match is: " + findMatch(24000, $postTimes); //23500

function findMatch($needle, $haystack) {
   $closest = null;
   foreach ($haystack as $element) {
      if ($closest === null || abs($needle - $closest) > abs($element - $needle)) {
         $closest = $element;
      }
   }
   return $closest;
}
?>

Finally you just need to implement:

 getPostTimesAsSeconds($postArray); //foreach converting dates to seconds-array

and

 pickPostBySecondsOfTheDay(23500); //foreach, finding the post matching the seconds of the day. 

Try the below example, file_get_contents is used for this example to fetch the xml. It will fetch all the feeds from the past 8 hours. Try to use DOMDocument to handle the xml feed and Datetime to manage the time comparisons needed.

$hour_interval = 8;

$feeds = file_get_contents("http://www.huffingtonpost.com/section/front-page/feed");
$doc = new DOMDocument();
$doc->loadXML($feeds);
$items = $doc->getElementsByTagName('item');

$today = new DateTime("now",new DateTimeZone("-04:00")); // do mind the timezone it is the one set in the xml feeds so it is needed for correct time comparison
$nowTimestamp = $today->getTimestamp();
$today->modify('-'.$hour_interval.' hour');
$eightHoursBeforeTimestamp = $today->getTimestamp();

$lastEightHoursItems = [];

foreach ($items as $item) {
    $pubDate = $item->getElementsByTagName('pubDate')[0]->nodeValue;
    $feedDate = new DateTime($pubDate);
    $feedTimestamp = $feedDate->getTimestamp();
    if($feedTimestamp<=$nowTimestamp and $feedTimestamp>=$eightHoursBeforeTimestamp) {
        array_push($lastEightHoursItems,$item);
    }
}
$random_keys = array_rand($lastEightHoursItems,3);
$c = count($random_keys);
for($i=0;$i<$c;$i++) {
    echo $lastEightHoursItems[$random_keys[$i]]->getElementsByTagName('title')[0]->nodeValue;
    echo $lastEightHoursItems[$random_keys[$i]]->getElementsByTagName('link')[0]->nodeValue;
    echo $lastEightHoursItems[$random_keys[$i]]->getElementsByTagName('description')[0]->nodeValue;
    echo $lastEightHoursItems[$random_keys[$i]]->getElementsByTagName('pubDate')[0]->nodeValue;
}