Exclude specific tags (by id) from the_tags

Within a single post I show all tags of the specific post by using this:

the_tags( $tags_opening, ' ', $tags_ending );

But sometimes I mark a post with a tag that should not appear on that single page (but this tag shall still be found on – for example – an overview page with all tags).

How can I exclude specific tags by ID from this list of tags within the post (and just on the single page)?

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.