Given the Puritans' view of the forest, what atmosphere does Miller create by setting this scene in the woods?

As the first answer states, Miller does not actually set any of his scenes in the forest.  The play refers to action that has taken place before the first scene between Rev. Parris and Abigail when Abigail discusses being in the forest.  The film version of the play does include two scenes in the forest--the girls' dancing with Tituba and John and Abigail meeting one last time in the forest (this second scene is not part of the play at all).

The Puritans' view of the forest as the devil's earthly home was a common teaching and has often been satirized and portrayed in American Literature.  Washington Irving, who regularly mocked Puritan superstitions and traditions, uses the forest as the devil's home in "The Devil and Tom Walker." Nathaniel Hawthorne, relative of the Puritan leader, Judge Hathorne (from The Crucible), sets the "earthy" scenes between Dimmesdale and Hester in the forest for The Scarlet Letter.  So, as you can see, the Puritans believed that the darkness of the forest hid the sins or sin nature of mankind from the "light" of God and provided a perfect setting for the devil to roam with his black book looking for humans who were willing to sign away their souls to do the devil's bidding.