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You can create a function for your ajax request, and set a timeout so it repeats. You will also need to send the ids of the current news articles being displayed so you can exclude them from your new WP_Query in your PHP

UPDATED – This now uses a data attribute to create an array of post ids to exclude off the next ajax request.

live.php

<div class="latest-news">
      <?php 
        $current_posts_ids = array();
        $args = array(
          'post_type' => 'post',
          'post_status' => 'publish',
        );
        $my_posts = new WP_Query( $args );

        if ( $my_posts->have_posts() ) : ?>
          <div class="posts">
            <?php while ( $my_posts->have_posts() ) : $my_posts->the_post() ?>
              <?php array_push($current_posts_ids, get_the_ID() );?>
              <article data-id="<?php echo get_the_ID()?>">
                <h2><?php the_title() ?></h2>
                <p><?php the_excerpt() ?><p>
              </article>
            <?php endwhile ?>
          </div>
        <?php endif ?>
 </div>

 <script>
  $(document).ready(function(){

      var ajaxurl = "<?php echo admin_url( 'admin-ajax.php' ); ?>";   
      var dataArray =  <?php echo json_encode($current_posts_ids) ?>;

      function getMyPosts() {

            data = { action: "update_news", data: dataArray};
            $.post(ajaxurl, data, function(response) {
                // reset array
                dataArray = [];
                // append the new news posts                
                $('.latest-news .posts').append(response);
                // create new array of ids to exclude
                $('article').each(function(){
                    dataArray.push($(this).data('id'));  
                }); 
                // repeat function after 5 seconds
                setTimeout(function () {
                    getMyPosts();
                }, 5000)
            });

     }
     //run the function first time
     getMyPosts();   

});
</script>

function.php

add_action( 'wp_ajax_update_news', 'update_news' );
add_action('wp_ajax_nopriv_update_news', 'update_news');
function update_news() {
    global $wpdb;
    $data = $_POST['data'];

    $args = array(
        'post_type' => 'post',
        'post_status' => 'publish',
        'post__not_in' => $data,
    );
    $my_posts = new WP_Query( $args );

    if($my_posts->have_posts()) :
    while($my_posts->have_posts()) : $my_posts->the_post();          
        echo "<article data-id='".get_the_ID()."'>";
        echo "<h2>" . get_the_title() . "</h2>";
        echo "<p>" . get_the_excerpt() . "</p>";
        echo "</article>";
    endwhile;
    endif;

 wp_die();

}

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