Some cells can keep dividing (gamete-producing cells, epithelial cells, cancer, etc) but the majority of cells do not divide forever. I was talking about this somewhere with CoxRox, and the accompanying theory is quite interesting, and makes sense. The idea is that cells which do not have telomerase (a reverse transcriptase enzyme which rebuilds the chopped off bits of DNA after replication) can, once they've gone through replication enough times, run out of telomere DNA. Then, replication starts to cause actual coding DNA to be lopped off at the end, which would obviously lead to problems coding for whatever that bit codes for. Of course, in humans, this is happening to each end of each chromosome, so we lose a bit off each end of 23 chromosomes every single time we replicate. This is one idea as to why we get old/die.
There are also two reasons can think of off the top of my head, both from evolutionary biology.
First, there is the idea that any lethal genes are more likely to survive the later in the life cycle they are expressed. That is, if there is a lethal allele which kills at 10 years of age, it is not very likely to be passed on to any descendants. However, if a particular lethal gene expressed itself only at (specific for the point of simplicity) 35 years of age. At 35, and individual would have had ample opportunities to reproduce, and so this gene would be passed on. Of course, there can be many such genes, which may not be lethal, but can lead to some kind of deterioration individually. As an individual ages, more of these can "sum up," finally leading to death.
Second, there is the point that death is simply a necessary part of life. It is important in this case not to think of survival in the terms of an individual organism, but instead of the survival (more specifically, reproduction and passing on) of the organisms genetic material. Once the organism has reproduced, is there much selection pressure for the organism to keep living? Not really, since the genes involved are already passed on to the next generation. So, there would really be no reason for an animal to evolve a body which does not deteriorate with age? Evidently not, since its progeny are more numerous and younger, more able to get to resources, it would seem that the torch is passed to them in terms of passing on the genetic material.
There are also two reasons can think of off the top of my head, both from evolutionary biology.
First, there is the idea that any lethal genes are more likely to survive the later in the life cycle they are expressed. That is, if there is a lethal allele which kills at 10 years of age, it is not very likely to be passed on to any descendants. However, if a particular lethal gene expressed itself only at (specific for the point of simplicity) 35 years of age. At 35, and individual would have had ample opportunities to reproduce, and so this gene would be passed on. Of course, there can be many such genes, which may not be lethal, but can lead to some kind of deterioration individually. As an individual ages, more of these can "sum up," finally leading to death.
Second, there is the point that death is simply a necessary part of life. It is important in this case not to think of survival in the terms of an individual organism, but instead of the survival (more specifically, reproduction and passing on) of the organisms genetic material. Once the organism has reproduced, is there much selection pressure for the organism to keep living? Not really, since the genes involved are already passed on to the next generation. So, there would really be no reason for an animal to evolve a body which does not deteriorate with age? Evidently not, since its progeny are more numerous and younger, more able to get to resources, it would seem that the torch is passed to them in terms of passing on the genetic material.