Disable core, theme and plugin update checks and notices and remove the update page from nav

disable-updates.php

<?php

// Don't disable on dev
if ( ! defined( 'WP_DEBUG' ) || ! WP_DEBUG ) {

	// Disable core update checking
	add_filter( 'pre_site_transient_update_core', create_function( '$a', "return null;" ) );
	remove_action( 'admin_init', '_maybe_update_core' );
	remove_action( 'wp_version_check', 'wp_version_check' );

	// Remove the updates menu item
	function yell_remove_update_menu() {
		 remove_submenu_page( 'index.php', 'update-core.php' );
	}
	add_filter( 'admin_menu', 'yell_remove_update_menu' );

	// Disable plugin update checking
	remove_action( 'load-plugins.php', 'wp_update_plugins' );
	remove_action( 'load-update.php', 'wp_update_plugins' );
	remove_action( 'load-update-core.php', 'wp_update_plugins' );
	remove_action( 'admin_init', '_maybe_update_plugins' );
	remove_action( 'wp_update_plugins', 'wp_update_plugins' );
	add_filter( 'pre_site_transient_update_plugins', create_function( '$a', "return null;" ) );

	// Disable theme update checking
	remove_action( 'load-themes.php', 'wp_update_themes' );
	remove_action( 'load-update.php', 'wp_update_themes' );
	remove_action( 'load-update-core.php', 'wp_update_themes' );
	remove_action( 'admin_init', '_maybe_update_themes' );
	remove_action( 'wp_update_themes', 'wp_update_themes' );
	add_filter( 'pre_site_transient_update_themes', create_function( '$a', "return null;" ) );

}

Very nice; for WordPress 3.4+, you can now consider using __return_null here as well, instead of create_function().