How to add date to navigation bar

It would be better to hook into the wp_nav_menu_{$menu_name}_items filter and add your date item in there:

add_action( 'init', 'wpse_106781_add_menu_items_hook' );

function wpse_106781_add_menu_items_hook() {
    $theme_location  = 'thirdnav';
    $theme_locations = get_nav_menu_locations();

    /* Make sure the theme location is valid */
    if ( ! isset( $theme_locations[ $theme_location ] ) )
        return;

    /* Get the navigation menu assigned to that theme location */
    $menu = get_term( $theme_locations[ $theme_location ], 'nav_menu' );

    /* Make sure the menu is valid */
    if ( ! isset( $menu->slug ) )
        return;

    /* Add the filter hook */
    add_filter( "wp_nav_menu_{$menu->slug}_items", 'wpse_106781_thirdnav_menu_items' );
}

function wpse_106781_thirdnav_menu_items( $items ) {
    $items .= sprintf(
        '<li class="navmenu-date"><time datetime="%s">%s</time></li>',
        date( 'Y-m-d' ),
        date( get_option( 'date_format' ) )
    );
    return $items;
};

As this function affects the front-end appearance, it would be appropriate to place it in the theme’s functions.php file.

Not only will it add the current date to the thirdnav menu, it will also use the appropriate HTML5 <time> element to represent the date. You can use the wp_nav_menu() function in your templates as normal.

You can style this item by using the class that is between <li class=" and "><time – in this case it is thirdnav-date, but it can be any valid HTML class you choose.