Not many philosophers think about immortality. This is because even if you could somehow stop the aging process eventually something would kill you (a freak meteor accident for example). Of these few I know only one, Heidegger, and he thought that being immortal would drastically change people, to the point where we could no longer consider them people at all. Heidegger’s exact reasons for believing this are hard to understand, because of his atrocious method and style, but I think the gist of his idea is that people are motivated to act because of their impending deaths, and that without this motivation life would change drastically.
But before we even discuss what the consequences of immortality might be we need to take another look at the possibility of immortality. And whether immortality is possible depends largely on how we define what it means to say that the same person exists at different times. If we define it in terms of physical constitution (they are the same person because they are made up of basically the same stuff) then there is no possibility of immortality. Given long enough, say a few thousand years, any biological entity is going to endure some trauma that won’t be survivable, even with the best medicine. However, if we define what it means to be the same person in terms of psychological continuity (in terms of mental properties alone) then immortality might be possible. It would be necessary though to copy the mind of the person into a computer, and then make copies or distribute the processing so that no event could wipe out the critical information. Such technologies are certainly possible, and will probably be developed within the next thousand years (not anytime soon, we still don’t understand the mind well enough). In this case the kind of disaster that could wipe out a person would be so unlikely that the odds are that they would live until the end of the universe, or until there was no more usable energy in the universe. Which of course brings us to the other problem with immortality, that given enough time the universe itself will end, and take everything with it (either by collapsing or reaching a state of maximum entropy). However, I don’t see such considerations as having much of an impact on our thought about the effects immortality might have, since given such a long period of time the probability that the person will choose to no longer be immortal approaches one (especially since computers can be sped up, increasing the amount of subjective time available), and so for all practical purposes such people are immortal, in the sense that they can live as long as they wish.
So let us assume that we did have a community of immortal individuals. Would they simply sit around and do nothing? I think not, for several reasons. One is that many of the things people do bring them pleasure, and, even though they won’t be less happy by accomplishing whatever they have set out to do in the future, people would rather be happy sooner rather than later. Competition also would still remain as a motivation. Only one person can make a particular new discovery, and only one person can be the best in a given area. Even immortals are likely to want that one person to be them, and so will still be motivated by that desire. Finally there are a decent number of people currently living who believe themselves to be immortal (via some sort of afterlife). Now admittedly many of them aren’t sufficiently confident in this to really allow it to inform their actions, and so still strive to get things done now in the fear of not being able to accomplish them later. Still some people must truly believe, and thus really think that they have an infinite amount of time ahead of them. I don’t seem many of these people simply sitting around unmotivated, and thus true immortals are unlikely to act in that way either.
In fact I don’t think being immortal would change people much. Contrary to Heidegger, most people simply don’t plan their lives around their eventual demise, and so losing that motivation wouldn’t change much. Perhaps the only thing that would change is that people would be less focused on “making their mark” (often via having children), which would probably be a good thing (since very few people really have the chance to “make a mark”, and thus people end up dissatisfied with their lives).