If Twitter users agree to it, then it's not an invasion of privacy for tweets to be archived in the Library of Congress. By sending a tweet, you should be aware that you're sharing something with the public. If you're not prepared to accept the consequences of that, you shouldn't tweet. Having said that, I'm not sure what the point of archiving every single tweet is. That seems like a huge waste of government time and funding. There are certainly much more important things for the government to focus on.
No, tweets being archived to the Library of Congress is not an invasion of privacy. ANyone using any social media platform should go off the assumption that anything you post online is not subject to privacy laws. A private e-mail from one sender to another, maybe. A post of Twitter, Facebook, Instangram, et.al? Not so much.
For me once you go to the internet, all activities are going public. The Library of Congress’ planned digital archive of all public tweets holds great promise for the research community, providing long-term curation and access to this valuable information resource. Therefore, if you don't want your tweets archived for research, don't tweet!
Like any social-network, by default any thing posted to twitter is going to be public. Realistically a individual can not expect that their comments made on Twitter to be private. Given that "tweets" offer a snapshot of culture identity and norms, they will be valuable to future researchers when constructing a historical image of our century.