That said, if you are willing to run your own (local or cloud hosted, but with scripting) web or e-mail server with git repository, you could build this yourself using GitHub post-receive notifications (http://help.github.com/articles/post-receive-hooks) sent to your web server, or normal (without diffs) e-mail notifications sent to a special address on your e-mail server. The choice between web and e-mail notification would be made depending on where & how you are hosting your server - if you have good connectivity & reliability, web notifications will have lowest latency, but if your web-server is unreachable for any reason, you won't get a notification until the next commit that is made while your web-server is reachable; using e-mail will delay the whole process, but might be preferable if you are hosting this at home, and especially for any machine that is not always on.
It's a bit of scripting work, and requires that you have some hosting infrastructure to run the web or e-mail server, the local-pull script, and then the local post-receive hook to generate the actual e-mail with diffs, but it should be possible to set this up in less than a day.
I'd be happy to be proven wrong and delete my answer... but I think at the moment the answer is no. Not even the API for github Issues seems to allow this.
For your example, I think your best bet is to search for all labels except [bug]; or create a new label "non-bug". :)
You can of course contact github and put this "filter-out" feature in their suggestion box.
Best Answer
According to the help page it appears not possible.