The quick answer is that it depends entirely on the author's wishes. If George R. R. Martin does not want another writer to finish A Song of Ice and Fire if he should pass away before completing it, then that is exactly what should happen. Or, should not happen, as the case may be.
Of course, Robert Jordan and The Wheel of Time is an excellent example of the other side of this. The author knew he was terminally ill and made arrangements through his wife and editor, Harriet, that the series would be finished and that his fans would get to find out how the series ended. His wife selected another writer who was able to not only finish Jordan's work, but also do so in such a way that both honored and lived up to the standard that was set.
But, the author's wishes are paramount.
In that Mind Meld, Bradley Beaulieu had this to say,
Last year on Speculate!, the podcast I run with Greg Wilson, we were interviewing Scott Lynch about his wonderful” Gentlemen Bastard” series, and we got to talking about the implied contract writers create with readers—whether there was one, how far it extended; that sort of thing—and Scott said that he believed that the author owed the reader the full story. If you said you were going to provide a trilogy, you really do owe them a trilogy. I’m with Scott. Readers become very invested in their fiction (I know I certainly do), and I think it’s fair to say that if you put down good money for the first installment, you really ought to provide all the installments you said you were going to provide in the beginning. Now, Scott also said that you don’t owe the reader their version of the story, and I believe that to be true as well. A writer owes it to herself and the story to finish it the way she wants to.
I could not agree more. I understand the perspective of Neil Gaiman when he, famously, wrote that "George R. R. Martin is not your bitch." Which is to say that GRRM, or any other author, is not required to write on your schedule or write what you think they should write. This is true.
I agree with Bradley and Scott that there is an implied contract between readers and authors. When I buy "Book One of Potentially Awesome Fantasy Series" from Wonderful Author Person, I am buying it with the understanding that it is only the first volume in a series and that the series may take years or decades to complete. Obviously, I hope it will be finished sooner, but that is because I am an impatient little bastard. I understand that it may take a long time and that in some cases, the author may need to step aside and write something else while they are working to complete the series. That's the thing, though, I am buying "Book One of Potentially Awesome Fantasy Series" with the expectation that there will be "The Last Book in Totally Awesome Fantasy Series" to wrap things up.
There are different sorts of series and they carry different sorts of expectations. Naomi Novik's Temeraire novels are, for the most part, "The Continuing Adventures of Lawrence and Temeraire" and there is likely no true concluding novel because there can always be more adventures. This is the same with Scott Lynch's Gentleman Bastards. Sure, I know that there are seven books planned, but for the most part, they stand alone while building a larger tapestry. Or many detective novels and Vince Flynn and Tom Clancy. The Wheel of Time and A Song of Ice and Fire and Memory Sorrow Thorn and The Night's Dawn Trilogy are all different. They are telling a singular story and set the expectation of a final volume that wraps up the story the author intends to tell, regardless of of how the reader feels it should end.
If you knew going in that the author was not going to complete the sort of series which more requires an ending, would you have bought the "Book One of Potentially Awesome Fantasy Series" to begin with?
This all ties together with the point about legacies, I promise.
There are all sorts of reasons why an author does not finish a series and most of them a reader would have to be an unfeeling asshole to not empathize with or be able to understand. Sales might be weak and the publisher declined to finish the series. The author may have been ill. There may have been family issues which there is no good reason for us, the reader, to know about. As Neil Gaiman explained, the author may have been stuck or needed to recharge on different projects before returning to the promised book. Actually, this is how Brandon Sanderson stays so prolific. He switches up projects and works on different things so he can stay fresh for his longer works. It's just that Brandon writes so fast that we seldom have to wait very long. But imagine a slower writer who still needs time to recharge on different projects. The author may have died.
Bradley Beaulieu wonders "does the death of the writer absolve the author from that contract" and goes on to explain how, from his fan's perspective, it does not - but that it should be done carefully and with conditions. He later explains in the comments to the Mind Meld that as an author he agrees with those who believe the author gets to decide her or his legacy and whether a story will be finished by another writer.
The older I get, the farther I get from the fan's perspective and the closer I stick with "the author's wishes are paramount." As a younger fan, of course I wanted whatever series to be finished and would have said that it should be. Now, with a touch more maturity under the belt, my thought is that:
A) It sucks horribly that the author has passed away and all of my sympathy goes out to the author's family and friends and I hope that they all are able to find peace with the loss of their loved on.
B) I may never get to find out how this story that I loved ends and that's okay.
So that's where I stand. But I do have one caveat to all of this, and it is more of a wish than a hope. My wish is that if an author knows that she is nearing the end of her life, whether it is illness or age, that even if she doesn't want another writer to finish her work that she would be willing to pull together her notes and maybe sketch out how various story arcs are intended to resolve, and permit those notes to be released after her death - perhaps as a published volume for her estate or maybe just something online through her publisher / agent / friends. It would be a way to provide closure for the fans and readers who have spent years and money and emotion investing themselves in that author's work. The series may never be finished and the author never had the chance to finish it the way she intended, but here's a glimpse into how the various arcs and characters ended. It's not a demand and it isn't a requirement, but it would be something nice. It would be one last gift to the readers who have been following along for years.