Smithfield Foods Inc. will keep its Sioux Falls, S.D., pork plant closed indefinitely at the urging of the state’s governor, though the company’s chief executive warned of dire consequences for farmers and consumers.

South Dakota Gov. Kristi Noem on Saturday called on Smithfield to keep the plant closed after linking it to 238 cases of the new coronavirus, representing more than half of all cases in the county. Smithfield’s plant and other meatpacking facilities around the country have emerged as hot spots for Covid-19, the...