I quickly wrote this script based on a suggestion
It's a pretty simple script. All it does is move (or remove) the birthday notifications from your message centre.
After installing the script, it will just move the notifications down below your folders. But if you want to completely hide the notifications, you can go to Tools -> Greasemonkey -> User Script Commands -> Birthday Mover, and press that. You'll get a confirmation dialog asking if you want to move the notifications, or hide them completely. That setting will be saved so it always does what you picked. Just go back to that option if you want to change something.
You can install this script if you use Firefox
Copy this URL to a new Firefox tab:
For Chrome users, you can install this version. But it will just remove the notifications, not move it down:
Let me know what you think