Nope, the flickr_badge_beacon can't be the problem imo. I suggest you try this outside of silverstripe in a very basic/plain html page... just to find out if it's a SilverStripe <> Flickr issue, or if it's a general integration issue.
As far as I can tell, you wouldn't need the css either, you should see some loaded images in a plain html page even without the styles.

Someone over at smashing magazine said the problem might be with the mime types:

I took a look at the Flickr code and discovered that it's using document.write to dump the code on to the page... document.write does not work in XHTML. The reason it works on those other sites is that they are serving their XHTML files with the text/html mime (which puts it in HTML mode instead of XHTML mode) instead of the text/html mime as you are doing.

I am not exactly to sure what he is meaning but the website he is refering to other than mine is http://www.abduzeedo.com

Is there a way to change the mime types in Silverstripe? and is that what I am even meant to be doing?