Promises to pay at a future date are a dangerous tool in the hands of politicians and state employees, those who suffer little to no personal consequences when past promises are revealed to be based on faulty assumptions and thin air. Either someone ends up paying, usually the taxpayers, or the promises are broken. Pensions are, of course, just such a promise. The pensions industry in the US is a good example of the way in which entitlement schemes run awry even without any sort of external shock to the system, such as large numbers of pension recipients suddenly living 5-10 years longer than the models predict. This seems likely to happen in the relatively near future, given progress towards the effective targeting of mechanisms of aging by the research and medical development communities. There must and will be radical change in pensions, either willingly or otherwise.
If you work in social security, it's possible that your nightmares are full of undying elderly people who keep knocking on your door for pensions that you have no way of paying out. Tossing and turning in your bed, you beg for mercy, explaining that there's just too many old people who need pensions and not enough young people who could cover for it with their contributions; the money's just not there to sustain a social security system that, when it was conceived in the mid-1930s, didn't expect that many people would ever make it into their 80s and 90s.
When you wake up, you're relieved to realize that there can't be any such thing as people who have ever-worsening degenerative diseases yet never die from them, but that doesn't make your problem all that better; you still have quite a few old people, living longer than the pension system had anticipated, to pay pensions to, and the bad news is that in as little as about 30 years, the number of 65+ people worldwide will skyrocket to around 2.1 billion, growing faster than all younger groups put together. Where in the world is your institution going to find the budget?
Suppose for a moment that human aging never existed and that, barring accidents and communicable diseases, people went on living for centuries - their health, independence, and most importantly, ability to work, remaining pretty much constant over time. In a scenario like this, it's difficult to imagine why any government would go through the trouble of setting up a pension system that works the way the current one does. Paying out money to perfectly able-bodied people to do nothing for the rest of their lives just because they're over 65 would make no sense at all.
Thus retirement exists out of necessity more than desire. The health of average retirees doesn't interfere just with their ability to work but also to enjoy life in general. Most people over the age of 65 suffer two or more chronic illnesses; the risk of developing diabetes, cancer, cardiovascular diseases, dementia, and so on skyrockets with age. Many people imagine a longer, drawn-out old age in which ill health and the consequent medical expenses and pensions are extended accordingly, just as in the nightmares of social security planners. This is most definitely not what life extension is about, and it's obvious that extending old age as it is right now would not be a solution to the problem of pensions.
However, lifespan and healthspan - that is, the length of your life and the portion of life you spend in good health - are causally connected; you don't just drop dead because you're 80 or 90 irrespective of how healthy you are. The reason we tend to die at around those ages is that our bodies accumulate different kinds of damage in a stochastic fashion; as time goes by, the odds of developing diseases or conditions that eventually become fatal go higher and higher, even though which specific condition will kill you depends a lot on your genetics, lifestyle, and personal history. The idea behind life extension isn't to just "stretch" lifespan; rather, the idea is to extend healthspan, that is preserving young-adult-like good health well into your 80s or 90s, and the logical consequence of being perfectly healthy for longer is that you will also live for longer.
Again, the fundamental reason that pensions exist is to economically support people who are no longer able to do it themselves. We need to have such a system in place if we don't want to abandon older people to their fate. If life extension treatments take ill health and age-related disabilities out of the equation entirely, pensions as we know them today will no longer be needed, because you will be able to support yourself through your own work regardless of your age.