WordPress: Disable "Add New" on Custom Post Type

The combinations of the solutions above work in hiding the links (although someone could quite easily type the URL in directly.

The solution mentioned @3pepe3 relies on get_post_type() which will only work if there is already a post in the listing. If there are no posts, the function will not return anything, and the "Add New" link will be available. An alternative method:

function disable_new_posts() {
    // Hide sidebar link
    global $submenu;
    unset($submenu['edit.php?post_type=CUSTOM_POST_TYPE'][10]);

    // Hide link on listing page
    if (isset($_GET['post_type']) && $_GET['post_type'] == 'CUSTOM_POST_TYPE') {
        echo '<style type="text/css">
        #favorite-actions, .add-new-h2, .tablenav { display:none; }
        </style>';
    }
}
add_action('admin_menu', 'disable_new_posts');

EDIT: To prevent direct access if someone types the URL in themselves: https://wordpress.stackexchange.com/a/58292/6003

There is a meta capability create_posts that is documented here and is used by WordPress to check before inserting the various 'Add New' buttons and links. In your custom post type declaration, add capabilities (not to be confused with cap) and then set it to false as below.

register_post_type( 'custom_post_type_name', array(
  'capability_type' => 'post',
  'capabilities' => array(
    'create_posts' => false, // Removes support for the "Add New" function ( use 'do_not_allow' instead of false for multisite set ups )
  ),
  'map_meta_cap' => true, // Set to `false`, if users are not allowed to edit/delete existing posts
));

You'll probably want to set map_meta_cap to true as well. Without it, you won't be able to access the posts' editing pages anymore.