Using the Heartbeat API that shipped with WordPress 3.6, this sample plugin shows you how you can have your post list update for your visitors in real time when a new post is published. This code should be seen as inspiration, as it needs to be modified to work with the theme you're currently using.

index.php

<?php
/**
 * Sample theme index.php to be used with WG Posts Autoreload
 */
get_header();

if ( have_posts() )
{
    echo '<div id="posts">';

	while ( have_posts() )
	{
		if ( ! isset( $last_date ) )
			$last_date = get_the_time( 'Y-m-d H:i:s' );

		the_post();
		?>
		<article>
			<h1><?php the_title(); ?></h1>
			<?php the_content(); ?>
		</article>
		<?php
	}
	echo '</div>';
}
?>
<script type="text/javascript">
    var wg_latest_post_date = '<?php echo $last_date ?>';
</script>
<?php get_footer(); ?>

wg-posts-autoreload.js

!function ($, doc)
{
    $(doc).ready(function ()
	{
		// Set the interval to 5 secs, for debugging
		wp.heartbeat.interval('fast');

		// Tell the server when our latest post was published
		$(doc).on('heartbeat-send', function(e, data)
		{
            data['wg-posts-autoreload'] = wg_latest_post_date;
        });

		$(doc).on('heartbeat-tick.wg-posts-autoreload', function(event, data)
		{
			if (!data.hasOwnProperty('wg-posts-autoreload'))
				return;

			// OK, we got posts back. Let's add them to the list.
			$.each(data['wg-posts-autoreload'], function (index, post)
			{
				// This code is quite dumb.
				// This is where you create a new post DOM element
				// and inject it into your post list
				$('<article />')
					.append($('<h1 />').html(post['title']))
					.append(post['content'])
					.hide()
					.prependTo($('#posts'))
					.slideDown();

				// Update the variable that we send to the server
				wg_latest_post_date = post['date'];
			});
		});


	});

}(jQuery, document);

wg-posts-autoreload.php

<?php
/*
Plugin Name: WG Posts Autoreload
Description: An example plugin showing how you can utilize the Heartbeat API of WordPress 3.6 to have your post list automatically updated when a new post is published.
Version: 0.1
Author: Anders Lisspers, Webbgaraget
Author URI: http://www.webbgaraget.se
License: Use the code as you please

*/

add_action(
    'plugins_loaded',
	array( WG_Posts_Autoreload::get_instance(), 'plugin_setup' )
);

/**
 * Plugin demonstrating the use of Heartbeat API for
 * autoreloading your posts list when a new post arrives
 *
 */
class WG_Posts_Autoreload
{

	/**
	 * Singleton instance
	 *
	 * @see get_instance()
	 * @type WG_Posts_Autoreload
	 */
	protected static $instance = null;

	/**
	 * URL to this plugin's directory.
	 *
	 * @type string
	 */
	public $plugin_url = '';

	/**
	 * Get the instance of this class
	 *
	 * @return WG_Posts_Autoreload
	 */
	public static function get_instance()
	{
		null === self::$instance and self::$instance = new self;
		return self::$instance;
	}

	/**
	 * Constructor intentionally left blank
	 *
	 * @see plugin_setup()
	 */
	public function __construct() {}

	/**
	 * Sets our plugin up
	 * @wp-hook plugins_loaded
	 */
	public function plugin_setup()
	{
		$this->plugin_url = plugins_url( '/', __FILE__ );

		// On heartbeat from client
		add_action( 'heartbeat_received', array( $this, 'add_new_posts_to_heartbeat_response' ), 10, 2 );
		add_action( 'heartbeat_nopriv_received', array( $this, 'add_new_posts_to_heartbeat_response' ), 10, 2 );

		// Enqueue our JavaScript
		add_action( 'wp_enqueue_scripts', array( $this, 'enqueue_scripts' ) );
	}

	/**
	 * Enqueues our script
	 * @wp-hook wp_enqueue_scripts
	 */
	public function enqueue_scripts()
	{
		wp_enqueue_script(
			'wg-posts-autoreload',
			$this->plugin_url . 'wg-posts-autoreload.js',
			array( 'heartbeat', 'jquery' ), // We need both heartbeat and jQuery
			false,
			true // Output in the footer
		);
	}

	/**
	 * Check if there are any new posts and if so,
	 * add them to the heartbeat response
	 *
	 * @wp-hook heartbeat[_nopriv]_received
	 * @param   array $response The data being sent to the client
	 * @param   array $data     The data received from the client
	 * @return  array
	 */
	public function add_new_posts_to_heartbeat_response( $response, $data )
	{
		if ( ! isset( $data['wg-posts-autoreload'] ) )
			return $response; // No client of ours has requested new posts

		$posts = $this->get_posts_since_date( $data['wg-posts-autoreload'] );

		if ( count( $posts ) === 0 )
			return $response; // There were no new posts

		$new_posts_response = array();
		foreach ( $posts as $post )
		{
			// This is the data being sent back to the client.
			// The following code needs to be updated, along with the
			// JavaScript receiving it, to match your theme's post list data.
			$new_posts_response[] = array(
				'title'      => apply_filters( 'the_title', $post->post_title ),
				'content'    => apply_filters( 'the_content', $post->post_content ),
				'date'       => get_the_time( 'Y-m-d H:i:s', $post->ID ),
				'permalink'  => get_permalink( $post->ID ),
			);
		}

		// Return our data
		$response['wg-posts-autoreload'] = $new_posts_response;

		return $response;
	}

	/**
	 * Gets all posts that were published since the given date
	 * @param string $date Date on the format Y-m-d H:i:s
	 * @return array
	 */
	public function get_posts_since_date( $date )
	{
		// Filter the query to only grab posts newer than the specified date
		$since_date = create_function( '$where', 'return $where .= " AND post_date > \'' . $date . '\'";' );

		add_filter( 'posts_where', $since_date );
		$posts = get_posts( array( 'suppress_filters' => false ) );
		remove_filter( 'posts_where', $since_date );

		return $posts;
	}
}