Twitter officially launched in 2006, but Presidential tweets didn't come into vogue until after Barack Obama took office in 2008. Today, the account for @POTUS (and the equally blue-check-official @FLOTUS) posts regularly and carries millions of followers. And just like every other responsibility and privilege associated with the office of president, there's now an official transition plan in place for this, too.
Yesterday, the White House published "The Digital Transition: How the Presidential Transition Works in the Social Media Age," essentially outlining the plan for how to preserve the history of presidential social media while ensuring the accounts transition securely and smoothly to President Obama's successor (whether or not they choose to follow in his BlackBerry and then Android footsteps).
For the headlining Twitter accounts, all media posted by the Obama administration through @POTUS will transition to a new handle, @POTUS44. The same will happen for accounts like @FLOTUS, @PressSec, and @VP. The National Archives and Records Administration (NARA) will maintain these new "44" accounts to preserve the digital record. On January 20, 2017, the newly elected president and his or her administration will then receive access to the @POTUS et al accounts, which will all maintain their current follower counts.