The IRS announced that its Free File program is up and running as tax filing season nears, despite the government shutdown.

The Internal Revenue Service plans to issue refunds if the shutdown persists once tax filing season starts Jan. 28. But it isn’t clear how long it will take the agency to send taxpayers their refunds, much less assist them with any filing problems along the way. Much of the agency remains furloughed.

The program—a free, electronic filing software...