This is a twofold question:
What is the oldest known thing still used for its intended purpose, though the original is gone? For example, parts of the Appian Way or Pictish bog routes are still used as roads, and active religious institutions are often built in the same location as older ones, though I assume the original materials are buried or gone.
What is the oldest known actual object still in use? For example, if an active road still had Roman-era stones, or if an active temple were built using parts of an old one.
Not trying to be humorous, but many ancient tombs could be considered in use for their original purpose.
One issue I see with this question is that we're not entirely certain of the original purposes of many ancient structures, as many have been repurposed throughout history, for example, the Parthenon. Also, the interpretation of "intended purpose" could be pretty broad; there are still events held at the Colosseum, which was constructed as a venue of entertainment, and there are many ancient churches that have been converted to mosques - is "religious worship" the intended purpose meant, or would you mean specifically the same type of worship to the same religious tradition as originally constructed?
Mundeshwari Temple is believed to have been functioning as a Hindu temple without interruption since the second century and still hosts Hindu functions today.
The Tower of Hercules, in Spain, and referenced by sources in the 2nd century, still functions as a lighthouse.
Hodge, A. Trevor (1992), Roman Aqueducts & Water Supply references ancient dams and aqueducts, some that are still in use, such as Lake Homs Dam in Syria, though the current dam is expanded from the original build under Diocletian
What is the oldest known actual object still in use? For example, if an active road still had Roman-era stones, or if an active temple were built using parts of an old one.
The Temple Mount in Jerusalem is still supported by the Western Wall, which was built during the Herodian renovations of the Second Temple in the 1st century BCE. It was built as a retaining wall for the mount, which was a giant raised platform for the temple. While the Temple itself was destroyed in 70CE, not all the works surrounding it were--many were buried, or repurposed. While its modern use as a destination of Jewish pilgrimage is somewhat newer (by a few centuries at least), its original purpose was to be a retaining wall for the mount, and for that purpose it's working just fine.
http://en.wikipedia.org/wiki/Arkadiko_Bridge
The oldest bridge in the world. I'm not sure if this is the oldest thing still in use but it has to be close, so as of now, we are looking for something older than 3,000 years.
See also an older thread with a similar topic.