I'd like to play StarCraft 2 without seeing all the achievements and related messages. Is there any way I can permanently get rid of them?
You can't disable them without going offline. I'm sorry to say it looks like you're just going to have to suffer through the achievement system like the rest of us. Maybe you should consider getting as many achievements as possible so you either run out of them, or become so used to seeing the notifications that you start to automatically ignore them.