UPDATE, Original post at the end:

My original answer did not address the concern that the computers should still run their original, real, non-educational purpose.

I want to disagree with Fraxinus on the point that Voyager 2 is the oldest still running computer as Pioneer 6, 7 and 8 might still be running. Last contact was made in 2000 and since then has not been attempted again. There is in my opinion no reason to suspect that they are defunct since Radiation decreases by distance squared and that should be the most danger they could receive. It is also highly unlikely that all three of them have been taken out by asteroids or other small objects.

There are also Pioneer 10 & 11 to consider but Pioneer 10 is presumed dead and 11's antenna no longer points at Earth.

Edit: after reading this post on the SE SE the conclusion one has to reach is that these things are all dead

If these are all defunct, then yes Voyager 2 might be the oldest.

ORIGINAL POST:

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 definitely 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 though it isn't running.