For tires, I think it depends how inconvenient it is for you to get a flat. If you're racing, or commuting in places where it would be dangerous to stop and fix a flat, then you need to be able to anticipate your tire wear, and I don't really have any advice on that.

But for myself, I'll keep a tire on at least until I get the first flat. I don't know anyone who rides on roads that are so debris-free that their tread will be seriously worn before they get a puncture! If the tire has a lot of cuts, or if the rubber is obviously very thin, I'll replace the tire then; if not, I'll keep riding it until I get another flat or two. When the flats start coming closer together, I know the rubber's not deep enough to keep the sharp stuff out and it's time to replace it.