I expect it's a legal gray area that mostly works in part because most random forum posts are pseudonymous.
Why not? Forum posts are copyrighted at the moment of creation just like any other work, and while one certainly gives a license (implied or not) to the forum, there's no reason why that license would extend to the IA.
So perhaps a new IA that only indexes creative commons licensed sites might be in order?
IANAL, but basically yes. It's one reason why the IA respects robots.txt even retroactively. The reality is that, if something was posted publicly by a copyright holder and intended to be shared, the overwhelming majority of people/entities don't care that it's being archived somewhere but there's no particular exemption for something like the IA.
To play devils advocate against myself, I have made mistakes in the earlier days of the internet, that I am glad the archives failed to keep. I do understand that there is a need for privacy friendly user sites, but I am sceptical about what tools are allowed to actually perform this structure. Right now, the internet is a threat to the power that be, which is why we will see an ever increasing attempt to legislate it into the ground. If we allow government corruption to seep into the internet anymore than it already is, the real concern will be one of censorship and propaganda, and user privacy is less to do with publicly posting things you shouldnt, but more to do with the corporate/government merger and data sharing that is going on around us. Loopholes everywhere for suppressing dissidents.