This question has a background perception about evolution process: A final result: “We have done, finished, got our last and final shape (naked apes), we have stopped all our mutations, we have isolated ourselves from every kind of environmental and physical dependencies, but still evolution did not make us immortal.” I am afraid it does not work like that. DNA has been evolving within an environment where everything transforms. Just as human technology, DNA produces solutions against environment, and each species are different interpretation. This catching-up and adaptation require mutations in organic cells. If you look at from the perspective of DNA, life on this planet is pretty durable, it has been occupying this space more than 3 billions of years. Individual species- like ourselves-, are fashions of this relatively long process. When we consider the age of universe as 13.7 Billions of years, 3 billion is a serious amount of time.
It is obvious that DNA has been doing everything in its scope to stay longer on this planet, trying new organs: eye to capture light, ears for listening to environment, brain to redefine the collected information. Species born with a set of tools, and new generation develops better equipments, if conditions are suitable, or if necessities arise. Where are the first PCs of 1970s? Nowhere around; because it would have been a cumbersome task trying to fit new components inside their archaic cases. We produced new ones instead. But we used the knowledge we obtained from the previous generation: Our PCs have new components, but their keyboards, mouse, monitor and box (case) look like their grandmothers. Modifying the old is not an efficient option; that’s why every dynamic system (human technology, DNA, language, etc.) evolves, adapts through “new models”.
Yet we know that if our planet’s gravitational cushion is disturbed and we start to fall into sun, or pushed away from the sun, or hit by an earth size heavenly body, all these 3 billion year of evolution will simply "be dead". The simplest answer for your question could be “we are mortal, because everything in this universe is mortal, and we are no way an exception to the rule, we have no priority over anything in this universe, we are not that special.” As long as we are in this universe we will never be. All we can do is gaining some time. Imagine if our consciousness would continue on non-organic material; let’s say if we could continue our existence using sub-atomic level interactions, we would survive longer. Make an exact working software replica of your brain and design it being able to perform on any group of atoms, save your existence in different parts of the galaxy and beyond. Live as long as galaxies, billions of years. Your existence is still bound to some sort of hardware, and it will not be “immortal”, just relatively durable. At the end, entire universe will collapse, get cold and be “dead” one day.
Immortality, or having a bit more time in existence, should find its solution within this environment. So it should necessarily follow the surrounding rules first, and then produce new opportunities. There is no controller behind this operations, nobody or nothing (including evolution, life or DNA) decides on mortality or immortality. If there was a “decision maker” behind life, we could have asked to him/her/it that “why aren’t we immortal?” Now where are you going to ask this question? To a cell, to soil, or to atmosphere? They would probably answer like that: “You are not alone matey, get over!”
That’s why we do not discuss our immortality, longevity issues with nature or with gods. We just try to do it within our own scope. Yet if we live millions of years without dying, our life in millionth year would have no sentimental relation with our life in 100th year for example. So many experience and modifications would make a totally unrecognisable –maybe traceable though- changes to our existence. “I was an ape a million year ago, now I am travelling around galaxy centre inside a giant star” What’s the existential relevance? Nothing. So everything necessarily dies, even if they are immortal. “I am going to keep my memories” Where are you going to store your experiences, on empty planets. You will give up one day. You will not be yourself anyway.
And how do you know that we are not the part of evolution’s search (or options) for immortality? Process has never stopped in this universe, and it will continue until the end of time.