Exclude most recent posts from specific category in wp_query()

Question

I have a featured posts slider, which uses the 'featured' tag. I have a query for this which grabs the 3 most recent posts with this tag and shows them in the slider. Below this, I have a list of 10 or so most recent posts across all categories. How would I write a query to exclude the 3 most recent posts with the 'featured' tag, as these are already in the slider (don't want post duplication), but then when they drop off the slider (i.e they're now the 4th most recent 'featured' post etc), they appear in the post list?

EDIT: Here is my code - I've written a shortcode so I can drop it into any page/widget/etc I want but the query for getting the 3 most recent 'featured' posts is here:

posts_per_page=3&post_type=post&tag=featured

and the query which grabs the 10 most recent posts across all categories is:

posts_per_page=10&post_type=post

Hope that's helpful :)

Thanks :)

EDIT #2: Used @Chip Bennet's answer, inside my short code which almost works but it only excludes one of the featured posts (most recent I think), not all 3 that are on the slider. Here is the code:

function loopShortcode($atts) {

    $excludeFeaturedPosts  = array(
        'tag' => 'featured',
        'posts_per_page' => 3
    );

    $featured_posts = new WP_Query($excludeFeaturedPosts);

    $featured_post_ids = array();
    foreach ( $featured_posts as $featured_post ) {
        $featured_post_ids[] = $featured_post->ID;
    }

    wp_reset_query();

    // Defaults
    extract(shortcode_atts(array(
        "posts_per_page" => '', // query parameters are defined by the user when you add the shortcode into the page
        "post_type" => ''
    ), $atts));

    $query_args = array(
        "posts_per_page" => $posts_per_page,
        "post_type" => $post_type,
        "post__not_in" => $featured_post_ids
    );

    $listedPosts = new WP_Query($query_args);

    // rest of short code content, e.g loop, output etc

    return $output;
    wp_reset_query();
} // end shortcode function

Answer 1

Assuming that the context is the Site Front Page (e.g. a custom front-page template file or custom page template), and not the Blog Posts Index or other archive index pages, the best approach would be to grab the post IDs when you create your slider query, and then specifically exclude them from your custom latest-posts query:

// Slider args
$slider_posts_args = array(
    'tag' => 'featured',
    'posts_per_page' => 3
);
// Slider query
$slider_posts = new WP_Query( $slider_posts_args );
// Grab post IDs
$slider_post_ids = array();
foreach ( $slider_posts as $slider_post ) {
    $slider_post_ids[] = $slider_post->ID;
}

// Recent posts args
$recent_posts_args = array(
    'posts_per_page' => 10,
    'post__not_in' => $slider_post_ids
);
// Recent posts query
$recent_posts = new WP_Query( $recent_posts_args );

If this were the Blog Posts Index, you would need to filter the main query at pre_get_posts, and exclude the posts added to the slider:

function wpse121428_pre_get_posts( $query ) {

    // Blog posts index main query
    if ( is_home() && $query->is_main_query() ) {

        // Slider args
        $slider_posts_args = array(
            'tag' => 'featured',
            'posts_per_page' => 3
        );

        // Slider query
        $slider_posts = new WP_Query( $slider_posts_args );

        // Grab post IDs
        $slider_post_ids = array();
        foreach ( $slider_posts as $slider_post ) {
            $slider_post_ids[] = $slider_post->ID;
        }

        // Remove slider posts from main query
        $query->set( 'post__not_in', $slider_post_ids );
    }
}
add_action( 'pre_get_posts', 'wpse121428_pre_get_posts' );