Simply put, we know that demons tend to flash their 'evil eyes' in a few situations:
- They are working to terrify a 'normal' person
- They are using or preparing to use a significant amount of power
- They are confronted by something they fear or respect, but did not anticipate
- They are feeling strong emotion
- They are trying to remind Sam or Dean that "Hey, I'm a demon." This could be because they're not being taken seriously, or because they are working with the Winchesters (for the moment) and some aspect of the plan is being dismissed because it would be impossible (for a human).
Given the on-screen evidence we have, and the fact that most (if not all) uses of the demon eyes occur in situations which match the above, my belief is that this is the 'normal' state of a possessed human, which the demon can disguise with a minor effort, similar to a human tensing a muscle.
It is something demons can consciously control, so they can change their eyes at will (for theatrics). It is also something they might drop when surprised or scared. Finally, when preparing to do something which will require all of their focus, they could relax it so they aren't splitting their attention.
Until one of the show's creators speaks to this or an episode references it, we won't have a complete canon answer, but this seems like a good 'best guess' response until then.