What should be happening, according to @Jack, is that Twitter is replaying a series of actions from a snapshot it took at 9AM, transferring @POTUS followers to the new @POTUS44 account where Barack Obama's history is archived and the new, fresh @POTUS as well. This was mentioned in the transition plans back in October, explaining that both accounts would end up with the (then 11 million or so) followers. The problem is that if you've unfollowed (or even blocked) since, it will apparently take time to catch up, and hasn't completed yet.
It's unclear if everything is working exactly as planned, but apparently the process is still ongoing. As it stands, if you'd like to follow or unfollow @POTUS, @FLOTUS or any of the other top-level White House accounts and took action on that within the last day or so, it may be better to wait until tomorrow to see how it shakes out. Or, you could keep flipping the switch and see what happens.
Either way, this is the first transition of the social media era, and it doesn't appear to be living up to user expectations. We'll see if things are done in a similar way the next time there's a change in the Oval Office, but for now, if there are any other updates or explanations we will update when they come in.
Update: The @Support account tweets that the problem "is being resolved."
Update 1/21: Twitter chief Jack Dorsey has explained what happened in a flurry of tweets. Anyone who followed @POTUS44 after 12PM Eastern were accidentally set to follow @POTUS as well, and those who had unfollowed @POTUS were suddenly marked to follow it. Other official accounts (like @VP, @PressSec and @WhiteHouse) were affected, too. Only about 560,000 users are likely to have run into this problem, but that's clearly enough to have caused a ruckus.