Maybe I overlooked something but I would say your first problem is you're targeting the wrong element:
You give your like an id of post_', $id ,'_likes
and then try to update article_'+post_id+'_likes
$('#article_'+post_id+'_likes').text(data);
Aside that, concatenation in php is done with . not , so I'm surprised this is working at all. A line like this looks suspicious to me:
<?php echo '<span class="like_button" id="post_', $id ,'_likes">', $likes, '</span>'?>
Why not just do this:
<span class="like_button" id="post_<?php echo $id; ?>_likes"><?php echo $likes; ?></span>