I don’t believe they do.

I have a face I don’t use very often, which has the built-in complication that shows the day of the month.

When I switch to this face, it always shows the day of the month when I last used that face, for about half a second, before it updates to today’s date.

Because of this, I don’t believe that the complication runs when the face isn’t showing.

This is an Apple app. It normally appears when you’re on a beta (either developer or public). If you’re not on a beta then I’m not sure why it’s there.