Without more context it’s hard to be sure, but it looks like this function is being called twice: once for each link ($direction=”next” and $direction=”previous”). The $posting parameter looks to be the next or previous post. If my assumptions are correct, you need to get the thumbnail for that post:
$thumbnail = get_the_post_thumbnail( $posting->ID );
and then insert it into the $post_navigation string being created. I would put it at the end of line 8 and see how that works:
$post_navigation .= '<a class="post-navi-' . $direction . ' clearfix" href="' . get_permalink( $posting->ID ) . '">' . $thumbnail .
Note that get_the_post_thumbnail() will return an image sized to the ‘post-thumbnail’ dimensions, and will likely be too large. So you may need to define a new size and pass that as a parameter to get_the_post_thumbnail(). You will also probably need CSS to style the link to accommodate the image.