You would need to add a counter and you would need to pass that counter to the template part. Passing arguments to the template part is a relatively new feature of get_template_part()
<?php
$counter = 1;
foreach ($featured_posts as $post) {
setup_postdata($post);
$args = array('counter' => $counter);
get_template_part('template-parts/content-archive', NULL, $args);
$counter++;
}
wp_reset_postdata();
?>
