Exclude specific tags (by id) from the_tags

Not sure that this is the best decision, but you can use the filter to cut your special tag without modifying templates.

add_filter('the_tags', 'wpse_320497_the_tags');

function wpse_320497_the_tags($tags) {
    $exclude_tag = 'word';

    // You can add your own conditions here
    // For example, exclude specific tag only for posts or custom taxonomy
    if(!is_admin() && is_singular()) {
        $tags = preg_replace("~,\s+<a[^>]+>{$exclude_tag}</a>~is", "", $tags);
    }

    return $tags;
});

If you confused with regexp, you can rewrite it with explode and strpos functions.

Hope it helps.