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We might hear, in olde-worlde speech, of mistreating old gents by means of water and fire; also a Shakespearean association -- the hero / villain's sylvan discomfiture.
You're on the right track wordplay-wise for part of the name (the Shakespeare-relevant bit -- though per answer, that relevance is more direct); but you're very much in the wrong part of our island.
I think you'll find that this station's name is officially and by convention, "the other way around" -- Dunkeld & Birnam. That said; you've got the right one. "Eld" -- archaic word denoting among other things, one or more aged human male(s). The mistreatment concerned: dunking the old geezers, and burning them. The Shakespeare bit: in the play it's foretold to Macbeth, I think by the witches, that things will go well for him "until Birnam Wood comes to Dunsinane".
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