At a recent Reddit AMA, Eric Lander, a professor of biology who played an important part in the Human Genome Project, answered this question:

Do you think immortatility is technically possible for human beings?

His response:

I don't think immortality is technically possible -- evolution has installed many many mechanisms to ensure that organisms die and make room for the next generation. I bet it is going to be very hard to completely overcome all these mechanisms.

This seems to me, at first blush, to exhibit the Evolution of Species Fairy fallacy. Evolution doesn't work to benefit species, populations, or the "next generation". If a mutation arises that increases longevity, and has no other downsides, then animals with that mutation should become more common in the gene pool, because they die less often. I remember reading that the effect would not be very strong, because most animals don't die of old age. But why would there be the opposite effect?

I am loath to attribute a very basic error to a distinguished professor of biology. Is there another explanation? Is the claim that evolution selects for mortality true?

Note: Eric went on to add:

I'm also not convinced immortality is such a good idea. A lot of human progress depends on having a new generation with new ideas. Immortality may equal stagnation.

This seems to be blatant rationalization of a preconceived idea that death is good. (I doubt he truly believes that extra progress is worth everybody dying.) So perhaps his first statement is also a form of rationalization. But it seems improbable to me that he would make such a statement about biology if he didn't think it well-founded. More likely there's something I'm misunderstanding.

ETA: one of the first Google results is this page at nature.com, The Evolution of Aging by Daniel Fabian, which goes into some depth on the subject. The bottom line is that it agrees with my expectation that evolution does not select for mortality. Choice quotes:

The Roman poet and philosopher Lucretius, for example, argued in his De Rerum Natura (On the Nature of Things) that aging and death are beneficial because they make room for the next generation (Bailey 1947), a view that persisted among biologists well into the 20th century. [...] 

A more parsimonious evolutionary explanation for the existence of aging therefore requires an explanation that is based on individual fitness and selection, not on group selection. This was understood in the 1940's and 1950's by three evolutionary biologists, J.B.S. Haldane, Peter B. Medawar and George C. Williams, who realized that aging does not evolve for the "good of the species". Instead, they argued, aging evolves because natural selection becomes inefficient at maintaining function (and fitness) at old age. Their ideas were later mathematically formalized by William D. Hamilton and Brian Charlesworth in the 1960's and 1970's, and today they are empirically well supported. Below we review these major evolutionary insights and the empirical evidence for why we grow old and die. 

How could a distinguished professor of biology, a leader of the HGP and advisor to the US President, get something so elementary wrong, when even a biology undergrad dropout like myself notices this seems wrong?

ETA #2: Gwern points to the Wikipedia article on Evolution of Ageing, which lists several competing theories of the evolution of aging (and therefore mortality). This shows the subject is more complex than I had thought and there may be good reason to believe mortality is selected for by evolution (or at least is reliably linked to something else that is selected). 

I should be glad that I didn't discover an obvious error being committed by a distinguished professional, even if he may be ultimately wrong!