In some places yes, some places no. For example, on the news feed, it doesn't work. But if you make a comment in a box you can copy/paste there. It seems that if you have a text box, it works. Another example. If you open a calender entry you can't copy/paste, but if you edit that calender entry, you can copy/paste there. However there are some places w/o a text box you can copy/paste. In IE, I can copy text and then paste it into an office document, or email, etc.
edit: sorry saw the twitter part of the question. In rowi, birdsong and twitt you can't copy directly from a tweet, but if you retweet, which brings up a text box, you can from there. Another interesting observation. I can copy from IE, but not from surfcube or 7 tab browser pro...
Facebook, I tried (only in the news feed) and no luck. You can still copy when typing ofcourse. Twitter, not sure. I use Birdsong (for Twitter) which had that functionality before NoDo was even out to the public.