これは、Reddit iframe の「src」を更新することで実現できます。
$('some-surrounding-div-here').find('iframe').attr('src','http://www.reddit.com/static/button/button2.html?width=51&url=http%3A%2F%2Fdomain.com%2Fpost-num');
ここでは、完全に機能する例を示します。
<html>
<head>
<script src="//ajax.googleapis.com/ajax/libs/jquery/1.9.1/jquery.min.js"></script>
<script type="text/javascript">
$(function(){
$('#refresh_reddit_button').click(function(){
$('#reddit_button_container').find('iframe').attr('src','http://www.reddit.com/static/button/button2.html?width=51&url=http%3A%2F%2Fdomain.com%2Fpost-num');
});
});
</script>
</head>
<body>
<input id="refresh_reddit_button" type="button" value="Refresh Reddit">
<div id="reddit_button_container">
<script type="text/javascript">
reddit_url = "http://domain.com/post-num";
</script>
<script type="text/javascript" src="http://www.reddit.com/static/button/button2.js">
</script>
</div>
</body>
</html>