When you say nothing happens, does the shortcode disappear when viewing the post/page on the frontend of your site or does it display it as you typed it?
If the plugin encounters any errors, it will output them as an HTML comment. Try viewing the HTML source to see what that says and paste the result here. Of course, if it includes your API key, remember to leave that part out!
I know you said you tried everything, but just to be sure... You did replace these X's with your actual API key, right? :)
[mailchimp_campaigns apikey="xxxxxxxxxx"]