Stephen is correct that the WITCH is the oldest digital computer design that is still running but it is restored and thus might not be considered as an original.

I think after that the oldest still working computer is the Zuse Z22/13 built in 1958 (or 1956) which is currently owned by the Karlsruher Institute of technology and on loan to a local museum "Zentrum für Kunst und Medien" Centre for Art and Media. 
It is definetly the oldest vacuum tube computer still running


There also is a Zuse Z4 on display at the German Museum in Munich, which they claim would still be operational tho it isn't running.