pre_get_posts with WP_Query to prevent posts from specific tags

There’s a couple problems I foresee here. I’m assuming the featured portion is a meta value of some sort. We can run a conditional that will run for all queries on the Front Page but it would override the meta_queries instead of append to them. The hook would look something like this:

/**
 * Modify Theme Queries
 *
 * @param WP_Query $query
 *
 * @return void
 */
function theme_pgp( $query ) {
    if( is_admin() ) {
        return;
    }

    if( $query->is_front_page() ) {

        if( false !== $query->get( 'post', false ) ) {
            $query->set( 'tag__not_in ' => array( 'Featured_Slug_ID' ) );
        }

    }
}

I don’t know what your post type slug is or what your featured meta key is so it’s a little difficult to create an exact exclusion query. Hopefully the above points you in the right direction though.