Tensions are escalating at food plants between essential workers who feel pressure to stay on the job through the coronavirus pandemic and their employers, who are striving to maintain the country’s food supply chain without interruption.

Union officials and worker advocates are fighting for unpaid leave and other accommodations for workers who fear contracting the virus or spreading it to family. Companies are pushing back, in some instances threatening to fire workers who don’t come in or battling unions to make sure factories...