How to add post meta in while loop?

You should use init action hook. From the codex: init is useful for intercepting $_GET or $_POST triggers.

The following code is just a sample and is untested:

function wpse283607_handle_submit()
{
    // check if form is POSTed and nonce is valid
    if ( 'POST' === $_SERVER['REQUEST_METHOD']
         && wp_verify_nonce( $_REQUEST['_wpnonce'], 'my-action' ) ) {
        if ( !empty( $_REQUEST['rank'] ) ) {
            $ranks = (array) $_REQUEST['rank'];
            foreach ( $ranks as $post_id => $rank ) {
                add_post_meta( intval( $post_id ), 'rank', intval( $rank ) );
            }
        }

        // redirect to the same page (to avoid re-submit form data on browser reload/refresh)
        wp_safe_redirect(
            remove_query_arg( [ '_wp_http_referer', '_wpnonce' ], wp_unslash( $_SERVER['REQUEST_URI'] ) )
        );
        exit;
    }
}

add_action( 'init', 'wpse283607_handle_submit' );

Then your <form> would look something like:

<form action="" method="post"><?
    wp_nonce_field( 'my-action' );
    /**
     * @var \WP_Query $the_query
     */
    if ( $the_query->have_posts() ) :
        $i = 1; // not sure how you generate your "rank" but just an example
        while ( $the_query->have_posts() ) : $the_query->the_post(); ?>
            <input type="text" name="rank[<?php echo $post->ID ?>]" value="<?php echo $i ?>" /><?
        $i++;
        endwhile;
        wp_reset_postdata();
    endif; ?>
    <input type="submit" name="submitBtn" value="Update" />
</form>