Answer

At the time the soul departs, there are two kinds of things that are said: 1. The confession, which is for the sick person, to atone for his deeds. 2. There are verses (“Hear, O Israel” and chapters of Psalms) that those present there recite at the time the soul departs.
The first is a matter of Jewish law, and the second is a custom. The custom is relevant in any condition of the sick person. The halakhic requirement, in principle, applies to the sick person himself. However, nowadays there are many halakhic decisors who say not to suggest to a sick person that he recite confession, because this may harm him psychologically and be detrimental to him. If he is unconscious, then seemingly this has no significance, since the confession is his and concerns his own deeds. Perhaps there is room to say it into his ear on the assumption that he hears and understands but simply cannot speak. But that is worth asking a doctor who understands what the different conditions mean (whether the patient can hear and the like), although I am not sure the doctors know.
In any case, it is clear that there is no obligation to say it when the patient is unconscious, and it is doubtful whether there is any value in it.