you can do that using JQuery autocomplete plugin
and once you have included all of the needed JS files just add this code after your new post form
$terms = get_terms("post_tag");
$tags="";
$count = count($terms);
if ( $count > 0 ){
foreach ( $terms as $term ) {
$tags .= '"'.$term->name.'", ';
}
$tags = substr($tags,0,-2);
}
echo ' <script>
JQuery("#post_tags").autocomplete(['.$tags.'], {
width: 320,
max: 4,
highlight: true,
multiple: true,
multipleSeparator: ", ",
scroll: true,
scrollHeight: 300
});
</script>';
note: Now this is good if you only have a small number of tags, but if you have hundreds or thousands of tags then using an ajax solution is a must.