Add exception for specific tag

Question

I'm using this piece of code to generate Related Posts by looking at the tags of posts. The problem is I have use a tag named Featured in order for a post to show up in a slideshow. So posts are showing up as related because of that matching Featured tag, even though they are not alike at all.

So is there an easy to to exclude the tag "Featured" in this code?

<?php $tags = get_the_tags(); ?>
<?php if($tags): ?>
<?php $related = get_related_posts($post->ID, $tags); ?>
<?php if($related->have_posts() && $related->found_posts >= 3 && get_option('pyre_related') == 'On'): ?>

I am guessing there needs to be a tag__not_in or something similar included but I just don't know PHP well enough to know the syntax.

UPDATED: I didn't end up using Chip Bennett's answer (which may have worked). I had to find the functions.php which was hidden in a special subfolder inside the theme. Using tag__not_in did not seem to work. It seem to always return the same posts (the most recent posts even if they shared no similar tags. Here is the entire get_related_posts function.

function get_related_posts($post_id, $tags = array()) {
    $query = new WP_Query();

    $post_types = get_post_types();
    unset($post_types['page'], $post_types['attachment'], $post_types['revision'], $post_types['nav_menu_item']);

    if($tags) {
        foreach($tags as $tag) {
            $tagsA[] = $tag->term_id;
        }
    }

    $args = wp_parse_args($args, array(
        'showposts' => 4,
        'post_type' => $post_types,
        'post__not_in' => array($post_id),
        'tag__in' => $tagsA,
        'tag__not_in' => 795,
        'ignore_sticky_posts' => 1,
    ));

    $query = new WP_Query($args);

    return $query;
}

Answer 1

The get_the_tags() function returns an array of objects. So, you can step through the this array, and unset the object for the "Featured" tag, perhaps like so:

<?php
$tags = get_the_tags();

foreach ( $tags as $tag_key => $tag_object ) {
    if ( 'featured' == $tag_object->slug ) {
        unset( $tags[$tag_key] );
    }
}
// Continue with the rest of the code, here
?>

Note: untested. Also, assumes that the slug for the "Featured" tag is featured.

Answer 2

            <?php 
            $tags = get_the_tags(); 
            foreach( $tags as $tag_key => $tag_object ) {
                if ( 'featured' == $tag_object->slug ) {
                    unset( $tags[$tag_key] );
                    }
                }
            if($tags):
            $related = get_related_posts($post->ID, $tags);
            if($related->have_posts() && $related->found_posts >= 3 && get_option('pyre_related') == 'On'): 
            ?>