The world most certainly will end, no later than about five billion years from now. Our sun is halfway through its life cycle and when it finally enters its Red Dwarf stage it will slowly expand to a diameter that will swallow everything inside the orbit of Mars.
However, the sun's temperature will rise slowly. So much sooner than that, about one billion years in the future, the earth will become so hot that all of its water will boil off. At that point the prospects for life are pretty dim. Perhaps a group of humans will manage to build a dome with refrigeration and a supply of water so they and an ecosystem with some representative species can survive considerably longer. But I can't imagine a technology being possible that will allow them to live inside the sun. Besides, even if they migrate to one of the outer planets or moons, that will be a pretty tough life. I'm not sure how much energy a red dwarf emits; there might not be enough solar radiation to build and power an ecosystem in one of those places.
So... at some point they will have to invent interstellar travel. Assuming relativity is valid and the lightspeed limitation is real, and a voyage across our galaxy will take more than a hundred thousand years, they'll need generation starships to go exploring, so they will pretty much have to invent that same technology they would use to survive on a boiling earth: a fully contained ecosystem that will keep operating indefinitely on omnivorous nuclear fusion reactors, refueling at every solar system.
If humanity dies off, I could be content with the hope that some day someone else will land here and discover the carefully preserved and recorded artifacts of our civilization. In the short run that will require establishing an archive on the moon because the weather and tectonic activity on earth will destroy anything we could build in no more than a few million years. For safety I suppose we should actually put it on one of the moons of the outer planets, so that Red Giant sun doesn't swallow it up, in case it takes the aliens more than five billion years to discover it. And of course a few backup copies in other places.
Better yet, send it out in a fleet of starships, manned or unmanned. Let it find the aliens instead of vice versa.
If you're talking about the planet literally being destroyed so there's no trace of our existence, then my answer is no, I would not like that at all. It's time to send those archive capsules to Europa and Titan! Or better yet, let's build those generation starships and get a few tribes of humans out there in the rest of the galaxy where whatever happens to earth won't affect them.
Or perhaps download ourselves into the computers of the future, so we won't be so fragile.
It seems like there are a variety of ways to beat the "end of the world" problem, as long as we have maybe one more century of technological progress before it happens. Oh yeah, and as long as we can get the world's governments to participate. The same guys who are ignoring global warming.