Once the characters in No Exit realized that they were placed there to torment each other, why didn't they refuse to do so? When Joseph says "eh bien, continuons" ("very well, let's get on with it") at the end, is he implicitly consenting to participate in this? What's the significance of his agreement (or at least lack of refusal) in Sartre's overall philosophy? In the context of the story (and Sartre's philosophy), could the characters possibly have refused and what might have happened if they had?
Note that the realization that the characters are their own tormentors is made clear quite early in the play:
INEZ: It's obvious what they're after— an economy of man-power— or devil-power, if you prefer. The same idea as in the cafeteria, where customers serve themselves.
ESTELLE: Whatever do you mean?
INEZ: I mean that each of us will act as torturer of the two others.
And their immediate reaction is, indeed, to refuse to participate:
GARCIN: No, I shall never be your torturer. I wish neither of you any harm, and I've no concern with you. None at all. So the solution's easy enough; each of us stays put in his or her corner and takes no notice of the others.
For the rest of the play, the characters are trying not to torment each other -- by ignoring each other, by trying to comfort one another, by laying bare their secrets. But always, there is something that keeps them from being able to hold their peace, that keeps them interacting and orbiting one round the other. Even if it's something so trivial as a burning desire for a looking glass.
An important implication made throughout the play is that these three characters were put together deliberately, precisely because this exact combination would be so self-perpetuating:
ESTELLE: ...But why, why should we of all people be put together?
GARCIN: A pure fluke, I should say. They lodge folks as they can, in the order of their coming. Why are you laughing?
INEZ: Because you amuse me with your "flukes. "As if they left anything to chance! But I suppose you've got to reassure yourself somehow.
ESTELLE: So it was all fixed up beforehand?
INEZ: Yes. And they've put us together deliberately.
INEZ: I know. And you're another trap. Do you think they haven't foreknown every word you say? And of course there's a whole nest of pitfalls that we can't see. Everything here's a booby-trap.
What this means is that the combination, the premise, is contrived rather than happenstance. Not any three individuals would torment each other eternally, not even any three damned sinners. But these three were specially picked and matched to each other because they would torment each other forever, and would be incapable of stopping themselves, even as they recognize what they are doing.
According to Sartre himself, they could have refused. In the many time misquoted "L'enfer, c'est les autres" ("Hell is other people"), Sartre made it clear that he did not imply people would always torture each other, but rather, that flawed relationship would always push people to hell, as there was nothing more important to us that the view of others.
Hence, the reason why the characters end up torturing each other would be that their relationships are flawed, at least according to the Comments on Huis Clos, 1964 that I reproduce here (translation is mine, if you find a more satisfactory one I would be please to edit it in).
“ Hell is others ” has always been misunderstood. It was thought that I wanted to infer that our relationships with others was always flawed, that they were infernal relationships. But, it was something totally different I want to say. I want to say that if our relationships with someone else are twisted, vicious, then the other is but hell. Why? Because others are, in the end, what matters most for ourselves, for our own understanding of who we are. When we reflect on ourselves, when we try to know ourselves, ultimately we use the knowledge others have gathered on us, we use the means others have used, have given us the possibility to use upon ourselves. Whatever I might say of myself, the judgement of the other is present in my speech. This implies that, if my relationships are flawed, I am putting myself at the hand of others and in this case, I am in hell. And there are a great many number of people in the world that are in hell because they depend too much on the judgement of others. But this doesn't mean at all that we cannot have relationships with others, it only marks the fundamental importance of each other on each of us.
— Sartre, Commentaire sur le CD Huis Clos, 1964