How successful are the playwrights in going against religion in the play Inherit the Wind written by Jerome Lawrence and Robert E. Lee?

I wouldn't say that Inherit the Wind is anti-religion as such. Rather it is an extended critique of a kind of fundamentalist religious mindset that seeks to tell people how and what they should think. It is not just Bertram Cates who is on trial; it's the whole town and its fundamentalist convictions. The strategy of Lawrence and Lee in critiquing religious fundamentalism is reflected in the character of Henry Drummond, the defense attorney. He gives Matthew Brady sufficient rope on the witness stand, so to speak, giving him ample opportunity to make himself and his literalist interpretation of the Bible look utterly ridiculous.
At the same time, Brady is presented as a generally decent man, albeit hopelessly ignorant when it comes to science and modern learning. It's notable, for example, that he is presented in a much more sympathetic light than the Reverend Brown, who not only opposes Cates but actually condemns anyone who seeks forgiveness for him. Brady, despite his hostility to Cates and his teachings, is still enough of a Christian to remind the congregation of the need for forgiveness.
The play's overall lack of hostility to religion per se can be seen once more towards the end. After Matthew Brady collapses and dies, Drummond is genuinely upset. He also puts the cynical reporter Hornbeck in his place for his animosity towards Brady and religion as a whole.