Okay, I figured it out. This is weird.
That 3D object is a sort of robotic device, and inside of it is a 3D lamp to illuminate the inside (which is not visible from this angle).
Well, the “Ray Shadow” button was active in the Shadow and Spot panel for that lamp, and that’s what’s casting a shadow (in 360 degrees around the robot, because the light is inside) on the plane, which is otherwise invisible.
If I unclick the “Ray Shadows” button and re-render, the plane is perfectly invisible, but the shadow of the robot (from the sun) shows up.
I only figured this out when I replaced the robot with a simple sphere and saw that that took care of the problem. So, then I knew it was something about the robot model that was causing this.
Sigh. What a waste of time. Thanks for the help everybody.