How do you remove an array element in a foreach loop?

foreach($display_related_tags as $key => $tag_name)
{
    if($tag_name == $found_tag['name'])
        unset($display_related_tags[$key];
}

A better solution is to use the array_filter function:

$display_related_tags =
    array_filter($display_related_tags, function($e) use($found_tag){
        return $e != $found_tag['name'];
    });

As the php documentation reads:

As foreach relies on the internal array pointer in PHP 5, changing it within the loop may lead to unexpected behavior.

In PHP 7, foreach does not use the internal array pointer.

If you also get the key, you can delete that item like this:

foreach ($display_related_tags as $key => $tag_name) {
    if($tag_name == $found_tag['name']) {
        unset($display_related_tags[$key]);
    }
}

Instead of doing foreach() loop on the array, it would be faster to use array_search() to find the proper key. On small arrays, I would go with foreach for better readibility, but for bigger arrays, or often executed code, this should be a bit more optimal:

$result=array_search($unwantedValue,$array,true);
if($result !== false) {
  unset($array[$result]);   
}

The strict comparsion operator !== is needed, because array_search() can return 0 as the index of the $unwantedValue.

Also, the above example will remove just the first value $unwantedValue, if the $unwantedValue can occur more then once in the $array, You should use array_keys(), to find all of them:

$result=array_keys($array,$unwantedValue,true)
foreach($result as $key) {
  unset($array[$key]);
}

Check http://php.net/manual/en/function.array-search.php for more information.