Harry--his motivation is the mystery. Harry is the antagonist, the one who causes conflict for the protagonist. The ambiguity in the story, that starts in medias res, makes it difficult to know Harry's motivation. On the one hand, Harry might have fallen asleep and dreamed the snake advancing across his torso. On the other hand, there might have been a snake that advanced, completely traumatized Harry, and departed--and it may or may not have been a krait. Some readers are of the opinion that Harry is playing a deliberate practical joke but this is not a widely held opinion and is inconsistent with textual evidence.
But he didn't move. He didn't even turn his head towards me, but I heard him say, 'Timber, Timber, come here.' He spoke slowly, whispering each word carefully, separately, ... 'Stop. Wait a moment, Timber.' I could hardly hear what he was saying. He seemed to be straining enormously to get the words out.
Timber Woods is the narrator and third-party observer of and witness to the incident with the snake. The question of his motives isn't a large one as he is presented as a trustworthy narrator who honestly conveys events without bias--other than his natural concern and worry. In other words, there is no reason presented in the text for Woods to make his friend look other than he is nor to make the doctor look other than he is.
Doctor Ganderbai is the protagonist of the story. This is indicated by such things as the attention given to describing the Doctor's equipment and medical manner. His motives are clearly revealed so that his transparency contrasts with Harry's ambiguity. Ganderbai is clearly a conscientious and caring man of medicine. His medical ethics include the ruse of keeping his patient, who is clearly in a dangerous situation, calm and optimistic even though there may be little real reason for calm or optimism. He is motivated by an equanimity and self-assurance that are a blend of inner character strengths and professional excellence.
Ganderbai reached the bed be looked down at Harry and smiled, confident and reassuring, nodding his head to tell Harry it was a simple matter and he was not to worry but just to leave it to Dr Ganderbai,