Wordpress - Remove the Featured Image Meta Box

I haven't had time to test this but this looks like it should work for you.

add_action('do_meta_boxes', 'remove_thumbnail_box');

function remove_thumbnail_box() {
    remove_meta_box( 'postimagediv','post','side' );
}

Check this for more info.

Edit: The main change here is that you need to attach the function to do_meta_boxes instead of admin_menu

The post thumbnail is added to a post type as something this post type supports. If you want to remove post thumbnail functionality from a post type, you can call remove_post_type_support(). Regular posts are also defined as custom post types, so it should work for them too.

add_action( 'init', 'wpse4936_init', 100 /* Something high, to make sure all post types are registered */ );
function wpse4936_init()
{
    remove_post_type_support( 'post', 'thumbnail' );
    // Or remove it for all registerd types
    foreach ( get_post_types() as $post_type ) {
        remove_post_type_support( $post_type, 'thumbnail' );
    }
}