CARACAS—Venezuela’s central bank filed a lawsuit Friday seeking to shut down a U.S.-based website that publishes the black-market value of its tanking currency, the bolivar, alleging the market is destabilizing its economy.

The suit accuses three people behind the DolarToday.com website of “cyber-terrorism,” as they allegedly artificially manipulate the country’s currency for political purposes with the site, while enriching themselves.

The...