As humans, death is ingrained into our culture. We are often told death is "Just another part of Life" which is ironic because death is defined of a lack of life, everyone will die.
Organized religion takes advantage of this, they say by joining them you can achieve an eternal afterlife, serving some form of higher power. In my opinion it is better to reign in Hell than to serve in Heaven.
As humans, yes, immortality should be allowed. The issue would not whether or not it was allowed, but who it was allowed to. For example, a woman who's life mission is to have as many babies as possible and cannot read nor write would not be described as an appropriate candidate for immortality if she has a "why should I" attitude that is unlikely to change,
It should only be granted to those who respect our planet and understand the importance of population control and the natural world should such a thing occur.
I think a human who has lived for thousands of years or more would be a very valuable source of knowledge and that is the key and it should be our aim to achieve this in our lifetimes.
If we ever find more planets to colonize then we will not need such a population cap.
When it is invented (if ever) I think it should be allowed. If a person wants to be immortal and pay for it it is their own choice. I think a lot of people would not do it though because they would have to witness everyone they love dying while they are still living.
Nothing is immortal, everything has a due date, from Stardust we came and to Stardust we'll return. From a religious standpoint, God will call you home when it's your time, you can't cheat death. From a economic standpoint having a well trained resource that lasts 20-30 years or longer is an even better return on the investment to train them. Imagine if the greatest minds could of remind great for another 100 years (Tesla, Einstein, and the likes.)
Last and probably biggest concern for most over-population! First the world is not even close to being over-populated and any source who says other wise is ill-informed, the lying Elite, or just plain stupid sheeple. Proof you say, simple we use tons of food for bio-fuel, waste more fresh water than we consume, and if both the last statements are wrong than we would see plagues and diseases with high human death tolls. Second if humans could live to 1,000 years old, most probably wouldn't want kids in the early years. Third if Einstein, Tesla, and equally qualified colleagues, didn't age and was still alive today we wouldn't even debate over-population because of the advances made by our most brilliant minds of yesterday interacting with our most brilliant minds of today while eagerly waiting to brainstorm with our most brilliant minds of tomorrow, instead of them crumbling to dust.
I could ramble on for days on this subject, but, will close with biological immortality has tons of pros and very few cons. Honestly the only real con in my eyes is the longer cells live the higher the risk they become cancerous. Any other cons I've heard or came to mind can easily be offset with probably no intervention.
Obviously, you can't be immortal in the literal sense. What you can do is extend lifespans indefinitely, heal injuries and remain forever young. I think this should be allowed. Or more, I'm going to do it either way.
As for the "watching people die" thing, that's just stupid. People we care about die all the time. Pets die even more often, as they have shorter lifespans. Eventually you'd adapt.
If not feasible right away then the main task of the government should be to build up adequate resources to provide everyone with biological immortality. This would pose many benefits including world peace. What's the point of war when yourself and your opponents are both immortal? Humanity would be able to focus its energies on enjoying immortality and spreading out to the stars and maybe slowly and subtly helping other alien species reach the same point of development.
If biological immortality is invented, I think it should be allowed, but that it should also have limitations. The world is already becoming overpopulated and people are consuming more than what the Earth has to offer. If biological immortality is invented, there needs to be rules that those who wish to have it must adhere to, such as the inability to procreate.
I believe that if it's possible, we should go for it. As things stand right now, our time in this Universe is very limited. I, for one, dread the day my consciousness ends and I cease to exist. Biological immortality, whether through DNA manipulation, nanobots, or a combination of both, is the future of the human race.
The problem is that you will watch others would probably can't afford it (its definitely going to cost something) die in front of you. imagine watching generation after generation die. for one thing age is caused by the sun and earths gravity. so unless someone finds a way to defy both of those its not going to happen