They were kept in the Ark of the Covenant, which disappeared with the Babylonian conquest of Jerusalem in the 6th century B.C. With the exception of a few latter-day conspiracy theories, the assumption is the gold on the Ark was melted down by the invaders. The tablets were probably smashed to smithereens before the Ark, or perhaps only its golden fittings, was hauled off to Babylon.
One survival rumor from the first century B.C., however, is a bit more credible than the rest. The priests were reported then to have carried it to Mount Nebo for concealment in a cave. No one has ever reported having found it, though, and those caves have been crawling with explorers for decades now. It's not likely to be there.