How do I pair iOS Remote with iTunes?
OK, I finally figured this out.

First I opened Remote and clicked "Add A Device".

Then in iTunes, I clicked the phone icon below the volume slider. This became a drop-down menu, and one of the items was a Remote icon. (This didn't work every time; I first had to disconnect and re-connect my phone to the wi-fi network.)



Finally, I was able to enter the code in iTunes, and it paired the devices.

I didn't need Home Sharing turned on.