is_paged() can be used to tell if you are on any page other than the first page of an archive. You could use it like this:
<?php if ( $archive_subtitle && ! is_paged() ) { ?>
<div class="archive-subtitle section-inner thin max-percentage intro-text"><?php echo wp_kses_post( wpautop( $archive_subtitle ) ); ?></div>
<?php } ?>