A COMPLICATION? Ancient Tablets Displayed in Jerusalem Fuel Looting Debate (AP). The article says that suspicion has been raised that the tablets may have been looted in the 1990s, but the owner says that they were originally acquired legally in the 1970s. Whatever the merits of this particular case (and I do not have sufficient information to form an opinion), the whole issue of what to do about looted antiquities is fraught and difficult.
Background on the Babylonian-Judean cuneiform tablets is here and links. Some old posts dealing with the question of scholars using unprovenanced antiquities are here and here. See also these two essays in Bible and Interpretation:
New Policy on Cuneiform Texts from Iraq (ASOR, November 2004)
Should Scholars Authenticate and Publish Unprovenanced Finds? (Eric M. Meyers, February 2005)
UPDATE: The author attribution for the 2004 Bible and Interpretation essay was originally incorrect. I have corrected it above. Apologies for the error.