This part of the story happens in Chapters 1 and 2 of the book.
In the end, what happens to Kemmerich's boots is that Muller gets them. He had been wanting them and asking for them. He and Paul eventually decide that it makes more sense for Muller to have them than for some hospital orderly to steal them after Kemmerich dies.
As for the doctors, the understand that Kemmerich is not going to live. Towards the end, in Chapter 2, they refuse to come to see him because there is no real point in it.
When we first meet Kemmerich in the opening chapters of All Quiet on the Western Front he is suffering with an amputated leg in an army field hospital. When his friends, including our narrator Paul Baumer, go to visit, it is evident that he will die from his wounds. Kemmerich owns a pair of beautiful, very functional boots, and with no ceremony or prevarication, Muller asks bluntly if he can have them. Kemmerich is reluctant to give them up, but when he does die Baumer collects his friend’s things and the boots are given to Muller.
This is an indication of the impersonality of war, of the way a soldier’s perceptions and priorities must be altered in order to survive—to cope with the situation in which he finds himself. This is highlighted by Maria’s comments on Muller’s desire for the boots. It is not that Muller is unsympathetic, or that he doesn’t care for his friend. Rather, “he merely sees things clearly… the boots are quite inappropriate to Kemmerich's circumstances, whereas Müller can make good use of them. Kemmerich will die; it is immaterial who gets them.” It is this sort of practicality and distance that must be adopted in the midst of war—any sentimentality leads to waste, any confrontation of one's emotions could have dire consequences, could be a distraction that a soldier cannot afford.
The doctors, too, are operating with limited resources and cannot attend to each patient with time and care; they can only do what they must. They are overworked and faced again and again with death and suffering and therefore must work through the day with mechanical efficiency. They must preserve their own sanity with detachment. When Baumer approaches a doctor as Kemmerich is dying, asking for help, the doctor replies, “How should I know anything about it, I've amputated five legs to-day.”
War turns men into bodies both before and after they have died; to the doctors, Kemmelich is just another sick body, taking up a bed they sorely need for more wounded. His is the seventeenth death of the day, one of countless amputees, and the hospital staff have jobs to do—too many jobs. Emotional detachment is the only way they can survive, leaving the mourning to those who have the time.