Where do tweets go to be archived and searchable for years to come?

The Library of Congress of course!

All public tweets dating back to March 2006 will soon become part of an archive maintained by the Library. It will be publicly accessible to researchers and library staff for non-commercial research and display. For more information check out this article: Library of Congress Tweets That It Will Acquire All Twitter Messages L. Wood

Updated (August 2018): As of January 1, 2018, the Library of Congress has decided to end its policy of collecting every single tweet.(Don’t worry though, your tweets from March 2006-December 2017 are still safely stored in the archives, and the Library of Congress will continue to seek out historically important posts.)

Read more here. And find the official statement by the Library of Congress here.