Civil disorder comes when you have more unhappy people than happy people. How much of a problem war weariness is depends on your luxuries, entertainment settings, etc.
One annoying thing to watch out for is that you don't get any warning when war weariness increases. So you need to get into the habit of checking your happiness every turn and make sure the balance hasn't shifted. The good thing is that if you look in the advisor and all cities have a favorable balance (at least as many happy as unhappy), then if something happens between turns to adjust your balance, your cities won't go into disorder right away: you get to notice it the following turn, and adjust for it (e.g., by boosting entertainment).
I wish, though, that at the end of turn it would warn you if you have any cities that are about to go into disorder if you don't do something.