Democratic U.S. senators led by Elizabeth Warren of Massachusetts urged a Trump administration labor board appointee to recuse himself from a case that could end up being a big victory for corporations trying to make it harder for workers to unionize.

The case before the National Labor Relations Board deals with the right of companies to restrict workers’ online organizing efforts. In a letter sent to NLRB Chairman John Ring, a Trump appointee, Senators Kirsten Gillibrand of New York, Cory...