SFC does put some content into the header, albeit heavily filtered and parsed and such. It runs the shortcode specifically so it can get the "final" version of the content for putting into meta tags for Facebook.
Basically, you're encountering the opposite problem here. This is a case where you *want* that shortcode to be parsed properly and put into the header. The alternative would be for it to put the [mingleforum] shortcode there, unprocessed, and thus have that shortcode show up on the resulting Facebook Page's shares or posts. Not a great user experience.
Note that if a post has an explicit excerpt, the latest version of SFC (0.999, still in beta) should use that instead (assuming I fixed that bug.. which is debatable ;-) ). So you might try recommending that for the Page where the forum is displayed for the user to give it an explicit excerpt which lacks the shortcode, perhaps one even describing the forum in detail.
However, the shortcode will *still* get processed at least once, even in this case. One of the other things SFC is looking for in a post is images. These get put into the meta for Facebook to find and use on shares. For this case, the excerpt won't do, since the excerpt probably won't have images from the post. So it processes the shortcodes in the content in case they're using some kind of gallery plugin or similar, then it looks specifically for html IMG tags in the resulting content. So basically you can't assume that the first time the shortcode is being processed is when the shortcode is actually going to be displayed. A shortcode should return the result every time, and if a theme does it wrong, then that's really on the theme author. Not a great solution, but the only one that works everywhere.
To unhook this on a specific page template or similar, you can do
remove_action('wp_head','sfc_base_meta'); as of SFC version 0.999, available here: http://plugins.svn.wordpress.org/simple-facebook-connect/branches/new-sfc/