Remove Post if Advanced Custom Field is checked to fix paging

Use WP_Query instead, and you can use the meta_key / meta_value parameters:

<?php
$paged = (get_query_var('paged')) ? get_query_var('paged') : 1;
$args = array(
    'cat' => 10,
'paged' => $paged,
'posts_per_page' => 12,
    'post_status' => 'any',
    'order' => 'ASC',
    'meta_key' => 'archived',
    'meta_value' => 'true'
);          
$posts = new WP_Query($args);
?>
<?php while($posts->have_posts()): $posts->the_post(); ?>
    <?php the_title(); ?>
<?php endwhile; ?>