The last we heard about the ark was with King Solomon:

“The LORD has kept the promise he made: I have succeeded David my father and now I sit on the throne of Israel, just as the LORD promised, and I have built the temple for the Name of the LORD, the God of Israel. I have provided a place there for the ark, in which is the covenant of the LORD that he made with our fathers when he brought them out of Egypt.” 1 Kings 8:20-21

Some claim the ark made it to Egypt or Babylon when enemies invaded and pillaged the temple in the Old Testament. The book of Daniel talks about some items stolen from the Jewish treasure, but that appears to be nothing more than gold goblets. 

Was the ark there during the time of Jesus?  No mention of it, but for the temple to be complete it would need to be in place.  If that's the case, then maybe the Romans burned it and desecrated it in 70 AD when they destroyed the temple.

The apparent resting place of the ark is in God’s possession today and it will be on display in heaven.

Then God’s temple in heaven was opened, and within his temple was seen the ark of his covenant. And there came flashes of lightning, rumblings, peals of thunder, an earthquake and a great hailstorm. Revelation 11:19

Some archaeologists have claimed to have found the Ark of the Covenant, but all such findings are a hoax. “Raiders of the Lost Ark” was only a movie.

If it were found, people would worship it instead of the God who the ark symbolizes. It is best left hidden.