If you even try to look up a username on Twitter in the archive.org, you’ll notice that it isn’t being archived, as well. So, if someone Tweets something, and they then delete that Tweet, it won’t be available in the Internet Archive. However, if you were to do the following search at Google:
You’ll notice that my tweets, for example are indexed in Google:
After you do that search, you will see a small drop-down arrow. Click that arrow and you will see that there is a cached version in many cases:
You can view the Google cached version of the Tweet if someone has removed or deleted their Tweet on Twitter. I have noticed that the cached versions of Tweets tend to stick around for a while. In fact, I have a Tweet that I Tweeted in June 2013, close to 9 months ago.
Although the archive.org site is not indexing Twitter.com, it appears that Twitter is not stopping them from indexing or archiving the site. I don’t see any records in the Twitter robots.txt file from disallowing them from crawling.