Given background app refresh isn't currently feasible I find myself leaving my iPhone on its face with lowest brightness to conserve battery and so I don't see distracting text.
Would it be possible to turn off the screen or display a black screen after a set time? While keeping the app in the foreground...
Very interesting, thanks! Unfortunately iOS does not allow third-party apps like MUDRammer to turn the screen off; that is only a system-level setting.
One thing you might try in combination with a lower screen brightness is to pick a Theme that uses black as a background color.