I think there are two likely possibilities.
First, the Babylonians took it and simply destroyed it for the gold etc.
The second and to me, most likely is the Priests knowing it would be taken, hid it. It probably will never be found or maybe has been found by Romans etc. who would have done the same thing as the Babylonians.
Now where it went from there....
The Ethiopians claim to have it, guarded by specially designated priest in a church in Askam, Ethiopia.
Ebed-melech the Ethiopian eunuch who rescued Jeremiah from the pit, took it to Ethiopia while disguised as a traveling merchant, so the Babylonians would not bother him..........................
https://en.wikipedia.org/wiki/Ebed-Melech
From Wiki:
Ebed-Melech (e-bed-me’-lek, Hebrew: Latin: Abdemelech) is mentioned in the Book of Jeremiah chapter 38 as an Ethiopian official at the palace of king Zedekiah of Judah during the Siege of Jerusalem. The name is translated as Servant of the King, and as such may not be his proper name but a hereditary title. The text relates that he was an Ethiopian (Jeremiah 38:7) and a eunuch. [1]
Ebed-Melech is notable for rescuing the prophet Jeremiah from the cistern into which he had been cast to his death (Jeremiah 38:4–13). Later Jeremiah relayed God’s message to him saying that he, Ebed-Melech, would “not fall by the sword” during the Fall of Jerusalem to the Babylonians (Jer 39:15–18) because he had put his trust in Him (God). According to some extra-biblical legends this extended to Ebed-Melech never dying, instead joining the small group of holy people who enter Heaven while still alive.
The issue with the ark is that if you desecrated it with unclean hands...the ark killed you!. It’s probably hidden some place if God himself has not taken it off the Earth.
Scripture says if you touch it, you instantly die.
Third, the Romans took it and destroyed it.
Most likely its long gone