http://forrst.com/posts/jquery_livesearch_for_twentyten_theme-tV3

//livesearch.js

jQuery(document).ready(function ($) {
    var contentCache, searched = false;
    jQuery('#searchform').submit(function () {
        var s = jQuery(this).find('#s').val();
        if (s.length == 0) {
            return;
        }
        var submit = $('#searchsubmit');
        submit.attr('disabled', false);
        var url = jQuery(this).attr('action') + '?s=' + encodeURIComponent(s) + '&action=search_ajax'
        jQuery.ajax({
            url: url,
            type: 'get',
            dataType: 'html',
            beforeSend: function () {

                submit.attr('disabled', true).fadeTo('slow', 0.5);
                document.body.style.cursor = 'wait';
                var load = '<div id="content" role="main"><h1 class="page-title">Searching...</h1></div>';
                jQuery('#container').empty().html(load);
            },
            success: function (data) {
                submit.attr('disabled', false).fadeTo('slow', 1);
                document.body.style.cursor = 'auto';
                jQuery('#container').empty().html(data);
                jQuery('#ajaxback').click(function () {
                    jQuery('#container').empty().html(contentCache);
                });

            }

        });
        return false;
    });
    var timer, currentKey;
    jQuery('#s').keyup(function () {
        clearTimeout(timer);
        timer = setTimeout(function () {
            var sInput = jQuery('#s');
            var s = sInput.val();
            if (s.length == 0) {
                if (searched) {
                    jQuery('#container').empty().html(contentCache);
                    sInput.focus();
                    //jQuery('#search-form span.processing').remove();
                    searched = false;
                }
                currentKey = s;
            } else {
                if (s != currentKey) {
                    if (!searched) {
                        contentCache = jQuery('#container')[0].innerHTML;
                        searched = true;
                    }
                    currentKey = s;
                    if (s != ' ') {
                        jQuery('#searchform').submit();
                        jQuery('#s').attr('value',''); 

                    }
                }
            }
        }, 800);
    });

});

//functions.php

add_action('init','ajax_live_search_js');

function ajax_live_search_js() {
    wp_enqueue_script('jquery');
   wp_register_script('ajax_live_search', get_bloginfo('stylesheet_directory') . '/js/2010.js', array('jquery'), '1.0');
    wp_enqueue_script('ajax_live_search');
}

function ajax_live_search_template() {
    if (is_search()) {
        include(dirname(__FILE__) . '/livesearch.php');
        exit;
    }
}

add_action('template_redirect','ajax_live_search_template');

//livesearch.php

<?php

if(!$_GET['action']) { 
    get_header();
    echo '<div id="container">'; 
}

else { $do_ajax =1;}

?>

<div id="content" role="main">

<?php if ( have_posts() ) : ?>

<h1 class="page-title"><?php printf( __( 'Search Results for: %s', 'twentyten' ), '<span>' . get_search_query() . '</span>'); ?></h1>

<?php get_template_part( 'loop', 'search' );?>

<?php else : ?>

<div id="post-0" class="post no-results not-found">

<h2 class="entry-title"><?php _e( 'Nothing Found', 'twentyten' ); ?></h2>

<div class="entry-content">

<p><?php _e( 'Sorry, but nothing matched your search criteria. Please try again with some different keywords.', 'twentyten' ); ?><?php if($do_ajax ==1){ echo'<a id ="ajaxback" href="javascript:void(0);" title="Back to last page">Go back to last page?</a>';} ?></p>
                    
</div><!-- .entry-content -->
</div><!-- #post-0 -->

<?php endif; ?>

</div><!-- #content -->

<?php
    if(!$_GET['action']) { 
        echo '</div><!-- #container -->';
        get_sidebar();
        get_footer(); 
}

?>