Aristotle approached the question of story and meaning in this way: Why is it, he asked, when we see a dead body in the street we have one reaction, but when we read of death in Homer, or see it in the theatre, we have another? ~ Robert McKee, Story
When a fictional character is killed, no one dies – and the readers and viewers know this, no matter how much they identify with the story. So what does this narrative device actually mean? Aristotle thought that the death of a character evokes emotions in the viewers that then allow the author to convince them of his narrative argument. He calls this "pathos". Following Aristoteles, the death of a character "means" that the audience becomes more susceptible to what the author wants them to believe (the moral of the story). But that is the author's perspective on what he can do with the death of a character, and does not explain how the audience experience and understand it.
To answer this question, we may have to first understand how the audience perceives the character.
A character is no more a human being than the Venus of Milo is a real woman. A character is ... a metaphor for human nature. ~ Robert McKee, Story
A character is a symbol that stands in for something that affects the reader or viewer: their deams and aspirations; or their fears. And thus the death of a symbol may symbolize that a dream will remain unfulfilled or a fear is overcome. Death in the media, I believe, is not final but transformative, and not real but symbolic, and this becomes most apparent in the journey to the underworld (read "death") of many mystical heroes. But that is only my opinion.
Is there research that tries to answer this question from the perspective of psychology?
Or, in the absence of such research, how would you study it?
In his comment, Steven Jeuris remarks, that a fictional character's death means that he dies in the fictional world. Let me elaborate on that.
Readers of fiction don't read like news readers. A newspaper reader reads to learn what has happened in the world he lives in, and (maybe) to understand why. Because these events (and their reasons) affect his own life and the decisions he has to make.
Readers of fiction, on the other hand, do not read to be informed. They read to experience emotions, to escape their own lives, to be inspired, and so on. The facts of the fictional world are irrelevant to their own lives, but the emotions they elicit, the ideas they generate, and the time spent away from the demands of their own lives are important to them.
If a neighbor, politician or artist dies in the real world, and I read about it, this affects my own life. My life has been changed (because a person is missing from it) or it will be changed (because politics will change). If a fictional character dies, nothing changes in my life.
What is changed, is the story.
And story logic is not like the chains of cause and effect (or chance) that govern the events in real life. In the real world, many (if not most) people die through forces that lie outside of their influence. They die of old age or are run over by drunk drivers. In the fictional world, everything happens by design, and that design is based on the story's premise and the characters actions. In a fictional world, no one dies by chance, and even old age has a meaning.
Very simplified, in a (classic type of) story a character achieves his goals, if he overcomes his deficits. If he refuses to grow, he fails. Death is the reward for moral failure. Think of the hero killing his antagonist or the first two knights being eaten by the dragon: these characters don't just die and then they are dead, but their death means that their way of life was wrong.
But of course stories are usually more complicated than that today. Characters aren't simply good or bad, they are "realistic", like real people. And the world they inhabit isn't as simply structured as that of a fairy tale, it is "realistic", like the real world. In Game of Thrones, characters die apparently like in the real world: despite being the good guys.
But still, it is not the real world. If the good guy dies in Game of Thrones, it is not because Game of Thrones is realistic and characters in its world die just like people in real life, but because it is not what we are used to from narrative conventions and it surprises us. The death of the good guy in Game of Thrones means that we cannot trust what we have learned about how stories work. It does not mean that someone died. It is sad and frustrating, because we wanted Ned Stark to succeed, but at the same time it is thrilling and satisfying, because it is novel.
These are just examples, and there will be disagreement over the interpretation of these stories, but I hope they serve to illustrate that the death of a fictional character does not mean (to the reader !) that this "person" has died in the fictional world. It means something different or on top of that to the reader (who knows, which is important, that no one has died !).
Of course, every death in fiction might be interpreted differently – depending on the specific story. But I think that it must be possible to generalize and find an underlying similarity or several "types of fictional deaths".
Finally, I think that this question does not belong on another site, because the study of the psychology of reading books or viewing movies (that is, the psychology of the audience) is the task of the psychologist just like advertising psychology (which also deals with how stories work on their audience).
So if you want, think of my question of belonging in advertising psychology: Does the clean shirt the housewife pulls from the washing mashine mean that the shirt has been cleaned? No. It means that the washing powder is better than other washing powders (or, if you are a cynic, that that is what the advertiser wants us to believe). The meaning of the clean shirt depends on the intent of the narrator and the circumstances of the narration (in an ad).
Similarly, the meaning of a fictional death will depend on the (reader perceived) intent of the narrator (who, by the way, is not the author) and the circumstances of the narration (a book, a tv show). But I don't want to do a (subjective) literary interpretation (like I did here), but learn how this has been studied or, if no such studies have been undertaken, study it with psychological methodology.
Hope this now makes a bit more sense.