I went and re-read the moderation intro here in the lobby and while I remember historically that the OP had to agree to a lobby move, I don't see that in there now. So, not real sure on that one. Better to err on the side of asking, IMO.
I don't recall the OP's consent being required for a thread to be moved into The Lobby, but I usually asked anyway to make sure that it was ok; I don't recall anyone actually refusing such consent.
However, I usually asked because:
1. The OP's "name" would be associated with that thread, at least in the message index.
2. If the OP declined, the impact of the thread would often be severely hurt without the original post.
3. There still may be people who want to walk gently around Prosper (e.g., still have big bucks invested) and thought it was ok to air one's opinions in a non-Googled forum, and they may dread even brief exposure in The Lobby.
But, no, I don't recall any consent rule for moving threads from Prosper - General Prosper Chat or Lenders - General to The Lobby.