Heinlein's novel 'Time Enough For Love' is one example, mostly hopeful. Harlan Ellison's 'I have no mouth and I must scream' on the other hand hit it hard as a horror story.   Greek myths had Tithonys who aged but could never die, and Prometheus whose immortality was exploited by the Gods to make possible a horrible punishment.  Neal Shusterman's 'Arc of a Scythe' series (Scythe, Thunderhead, The Toll, and Gleanings) explores a society that's eradicated death and subsequently lost track of the idea that death (and therefore life) are meaningful.  David Eddings' 'Belgariad' has a lot of characters who are immortal and just keep dealing with it, each in their own ways, as the days keep going by.  
There are a lot of opinions on what dealing with immortality would be like, which is consistent with the fact that people are different and each author projects their own reactions and responses into the work.  That probably says more about how each of those authors might handle immortality than about how people at large would handle it.
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