Life extension is not the problem. It's already been happening. Just look how the life expectancy has changed in the last few decades, or the last half century. The theories about ageing are numerous (the telomeres are just one of the things participating), it is a complex process. But without any doubt, the improvements in healthcare, the abundance of food in a large part of the world, and other factors have already been doing this. I've read theories, that we might have the potential of reaching 140 years. I don't find it unrealistic, unless our civilisation ends earlier.
The problem is not primarily there. The problem is radical extension of the time we can spend with our cognitive functions intact. If we could change this, a much bigger part of the population would be learning languages for ten or twenty years longer without any need for an overall longer life.
It's not just about the neurological conditions, such as the Alzheimer's disease, or other not that hard to describe issues. The mental health is gonna be a huge challenge. The issue is more and more important (gerontopsychiatry is one of the specialties with bright future), and the long life will affect us in many ways (such as allowing us to accumulate much more trauma, more toxicity, and giving us more time to present stuff we've got predispositions for).
But the whole idea "what would change, if we had much more time to learn languages" is certainly fascinating, and it has already been explored in literature. I love scifi and related genres, as a sandbox for such ideas. A typical old vampire from modern popular literature is likely to speak several languages (lots of time, and lots of motivation to move between countries at leastevery few decades), and one or two dead ones. Too sad they don't have many other people to enjoy the language with anymore. And it's not just the vampires, but other such individuals too.
But a whole society reaching such longevity, that might be different. I personally guess we'd have too much time for homogenisation. Everybody would have enough time to learn English, I suppose the International Broken English would be the prevailing dialect, so the other languages would die out sooner than their natives.
Any benevolent AI would just further speed up the homogenisation, I guess. Or, we could reach a situation from a scifi book the name of which I cannot remember anymore, where all the education was happening just by uploading a cassette inside the brain. A very economic and fast way to get trained workforce. However, it came with some downsides, caused by the inability of most people to learn new things on their own, to improve the old ideas, or to think differently from the rest. So, if we could just have a language chip implanted, I don't think it would be such an awesome thing for our civilisation, even though most employers would love that