Well, logically speaking, if it's first person, if you kill the narrator, it would be hard for the story to continue unless you change the perspective to someone new. Which.. defeats the purpose of first person in my opinion. Stories are told pass tense, which means that the person has to survive in order to tell the tale. They aren't usually written in the present. You could put some twist to it that at the end this person dies and the narration is from the person telling the story on their death bed or the likes...
Reflecting on all the stories/movies I have enjoyed, I don't recall a point where the narrator died when they played an active role in the story. Two that come to the top of my head currently... 300... the narrator was one of the warriors who was the only one to survive and was sent back to Sparta. Ends with him finishing the tale on the precipice of battle with a full army at his back.
We have sandlot... who told the story through the eyes of Smalls and ends with him in current day being a sports caster observing Benny steal home.
Even the Hobbit, was told of Bilbo recalling his tale of his journey, having the trilogy begin and end with him in modern day. Lord of the rings, was a story narrated from Frodo's perspective and end with him handing his writing to Sam before sailing.
So I would say yes, the narrator must survive to tell the tale. The only way I could see you writing the narrator die is if you pull something like in the titanic... The story was told from Rose's perspective. Ending with her in modern day/present day after having finished her story and poetically dying on the waters above where the Titanic sank.
EDIT: In response to your edit of the question, my answer is no, knowing the person narrating is going to live or die does not take away from the thrill of the story. The thrill is in the journey itself as I commented on Mark's post.
I go on a vacation, and I tell my friends about the white water rapids trip I took where the water threw me around narrowly dodging rocks, hitting a wave and being tossed out of the raft! GASP THEN! I was tumbling under the water in the current gasping for air as my head came above the choppy water every so often! TO MAKE IT WORSE! There was a fork in the river where one path took me towards a water fall and I had no way of controlling myself! Before I knew it my back smacked into a rock holding me in place in the river. I managed to grab onto a tree root and pull myself to safety and wait for my friends to come get me. That my friends is how I ended up all bruised and with a broken rib.
Obviously here in this made up vacation... we know the person is going to live.. they are telling it to our face! But it's the suspense and excitement that keeps you listening as they recall the story of how they nearly died and came home in a body cast. You are invested in it because hopefully this person is a friend you care about!
Point being, make your character where the audience is invested in them! Whether they die or not won't make people care about them if they aren't invested. A good character with a good story will make people invested and enjoy the thrill. Even if they know this person will live or die.
To further add, there has been several anime where it was pretty obvious from episode 1 that said character would die at the end. Even when you knew it was coming, that didn't change the fact that someone decided to chop onions on the last episode and make my eyes water. You are bonded to the story, to the characters and their tale is entertaining enough that even when you know the outcome, it still emotionally moves you.