Does the donor of the Sharingan eye have to die so that it could be used by another user, or could they just exchange their eyes so that they do not go blind over obtaining the Eternal Mangekyou Sharingan?
There is no evidence to suggest that the eye "donor" has to die to donate his eyes for awakening the recipient's eternal Mangekyo Sharingan. While it seems to be theoretically possible to exchange eyes, it just so happens that in both the known instances of activating the eternal Mangekyo Sharingan, the donor died shortly before or after the transplant. Hence, the idea was left unexplored.
Izuna gave his eyes to Madara when he (Izuna) suffered a life-threatening wound, but it seems that he could have given his eyes to Madara earlier if he wanted to. However, since Madara loved his brother, they probably did not even think about that option.
Itachi's eyes were transplanted to Sasuke after Itachi's death, but again, he perhaps could have given his eyes to Sasuke without dying. He could not do it as he was pursuing other plans.
Eternal Mangekyo Sharingan can only be obtained by receiving eyes from close relatives. It is relatively rare for one person to achieve the Mangekyo Sharigan, much less for two people from the same family. Moreover, the Uchiha clan massacre eliminated whatever little possibility of that happening, so the idea of exchanging eyes has been left unexplored.
It is also possible that the author did not think too deeply about exchanging eyes and other possibilities, given that there are only two known instances of awakening of the eternal Mangekyo Sharingan.
Note This answer is based on events covered up to Chapter 624.
Well, when Obito gave Kakashi his Sharingan eye, he was not dead. So I'd say that this would be possible. Also according to the wiki,
Apparently, if a living Uchiha gives away their Sharingan, they will still have a link to the detached eye, and will be able to "see" what the eye sees in some instances, though this seems to be limited to a certain range