Elves don't died of grief, I think. They fade. The relation between their fear(spirit) and their body (I don't remember the specific name Tolkien used) goes wrong. That's one of the consequences of the work of Melkor. When the Elves lose their body, they are summoned by Mandos. Still They are free to go or not. The Elves who refuse the call of Mandos, stay in ME and may become wicked spirit. It is told Sauron used these wicked spirit.
It is true that as long as you have the motivations and the will to live, you tend to live longer. If you still have interests( you are not disenchanted), life is attractive and you don't want to die. So Elves are nearly immortal if there are willing to live. Human nature is different. What I have said above still apply for humans, but one day it is time for humans to leave the living world. I think trying to postpone the death will only bring pain. Tolkien clearly criticize the Numenoreans who have tried to gain a few years. The ones who did it gained nothing.
It is clearly unfavourable to do so.
It seems that Humans are less affected by the weariness of life, unlike the Elves. Elves live with the memory of the past. Most Humans, because of the shortness of their life, live with an incredible energy. They don't have time to spare. It is the future who is important, not the past.

Here in this world, humans have to accept their destiny. Death is our destiny, at least it cannot be avoided. So why not accept death with calm and serenity?

Mordan