I'm not talking about the challenges of achieving immortality in the sense of medical advances that eliminate deaths due to disease, aging, and not-immediately-fatal accidents, but rather the challenges we'll face if we can achieve potential immortality.
Let's also forget (for the moment) the potential of our minds living forever as simulations or models in a sufficiently powerful computer. I'm also not talking about the SF-nal concept of cloning full duplicates of ourselves (complete with memories) as a backup or co-worker. I'm talking about living in our God-given human bodies for as long as we can or want.
If we only have accidents, suicide, and homicide as causes of death, our lifetimes would extend dramatically. According to the Wikipedia article "List of causes of death by rate for worldwide statistics," roughly 90% of deaths are caused by diseases, aging, and the like. Only 10% of deaths are due to accidental or intentional loss of life (and you wonder why "accidental death" insurance is relatively cheap?)
An overly simple extrapolation suggests that if we are "biologically" immortal, our average life span would be in the range of 500 to 1000 years. You still might get trapped in a falling building, struck by lightning, or murdered by someone for profit or revenge.
If we don't drop the birth rate by a factor of ten, we'd soon be faced with a population explosion problem of unprecedented (among humans) magnitude.
If we did manage to drop the birth rate, we'd be faced with unprecedented economic and social problems stemming from the growing fraction of retirees and working adults compared to children.
For one, our current society is largely based upon the idea that you work for 40 or 50 years, then retire. That simply won't work without proportional increases in the productivity of workers. Remember, in the big picture everyone is living off of the productivity of those who are currently working. There is no such thing as savings. Money is simply a way of accounting for the distribution of wealth. Investment only works by boosting the productivity of current workers.
If everyone retired after 50 years of working (lets assume that 90% of the population is retired or at least not building things or farming), then the 10% that are being currently productive have to work hard enough to feed, clothe, house, and entertain the other 90%. They might not appreciate the burden.
Several other challenges of biological immortality have already been explored in various science fiction stories. Examples include social factors such as marriage, families, and even memory. Is the human brain adequate to retain a millennia of memories? Is the human brain capable of learning new occupations, century after century? I fear that our ability to learn (as evidenced by a child's superior grasp of new languages) may not prove up to the challenge. Several SF authors posit "memory cleaning"--the erasure of unwanted, unneeded, or simply excessive memories.
Will women be able to extend conception beyond the age of 50? We'd likely have to grow fresh, new ovaries, too. But the likelihood of more than 2 children per lifetime puts an even greater burden on population. If the average woman limited herself to only 2 children per century, after a thousand year lifetime she would have 20 children and over a thousand direct descendents. How's that for a population growth problem? I suspect that the government will restrict childbearing to mortals--once you undergo life extension treatments, you won't be allowed to have children.
I do think that Niven is wrong: the ancient will not be much more graceful than the young. I know that my coordination has not improved as I age. I am more careful than in my youth, but that's because I'm not stupid and realize that I'm not immortal (a major failing of the teenage years).
One big question: will the prospect of biological immortality increase or decrease the value we place on life? Will we take more chances, or fewer? We'll have more to lose, yet likely a greater likelihood of having enjoyed a full life.