Take a story's temperature by studying its tone. Is it hopeful? Cynical? Snarky? Playful?
Camus definitely approaches The Plague didactically. Not that we love him any less for it. The difficult part about discussing tone with a novel like this lies in trying to separate the tone of the narrator from the tone of the author. The trick is to think about how Rieux and other characters come across. For example, Rieux thinks he’s being objective, but Camus has him so overemphasize this point that it becomes ironic; Rieux is certain of his objectivity, but we all know it’s bunk. Another example is Paneloux; Rieux may be indifferent to the priest’s beliefs, but the novel (and therefore Camus) is clearly condemning him.
Now that we’ve broken the barrier between the author and his character, let’s get back to this moralistic tone claim we made up there. What we mean to say is that Camus clearly intends to teach us a lesson with his novel. We are expected to conclude from The Plague that the world is irrational, that suffering is inevitable and senseless, and that we should take up arms against it anyway. But the point in talking about tone isn’t to figure out what lessons we’re being taught; it’s the very fact that we’re being taught lessons at all. Hats off, in this case, since it’s hard to be moralistic without getting preachy. Just ask Father Paneloux.