Question: Instead of discovering that earth was round, we find that we live on a infinite, flat "earth".

What are some cultural implications if we were to live in a world like that (disregarding any physical impossibilities, and that gravity, biological/ecological systems remain roughly the same as reality).

One interesting idea is that cities could be tens of thousands of kilometers from another and that one can only explore so much of the world in their lifetime.

EDIT: I don't mean flat as in geometrically flat, but flat in general. There would still be mountain ranges and oceans and forests and stuff. But there would be different and exotic ecological systems as you travel.

Answer: 
The world would become progressively more alien as you moved across it.

Unlike on Earth, where animals and humans can move across the entirety of the world given a few thousand/million/tens of millions of years, an infinite flat earth would always have somewhere that's far enough away that nothing from where you are has ever been there.

This would probably lead to a continuous cycle of mass extinctions brought about by animals migrating in from a place where they have evolved in a manner that allows them to out-compete their neighbors. Something like Europe could exist, with mammals and birds forming most of the terrestrial fauna, when a group of highly evolved velociraptors finishes ten million years of gradual migration and then decimates the entire local ecosystem.

At a great enough distance, *nothing* would have even a single common ancestor. You'd likely get great patches of land in which only single celled or incredibly basic forms of life exist, since this form of life evolves in abiotic conditions relatively quickly, but then takes billions of years to become multi-cellular. These stretches would separate islands of complex life, which would become increasingly complex and highly evolved at their centers.

Aside from the mass extinctions that would occur when these life-islands grew into each other, different life islands could also evolve their own sentient species. These species, again, wouldn't share any common ancestors, and could well be separated by hundreds of millions of miles. They would all be incredibly alien to one another, more different than humans are from plants or mushrooms, but they could eventually come into contact with one another. Physically reaching one another would take quite a bit of time, since they'd likely be separated by immense distances, which would be difficult to traverse, but across which radio waves could probably travel.
A lot of people ignored the question and delved into the rabbit hole of physical impossibilities given the physics of our known universe that were ruled out in the question. Given an infinite, planar type world with similar characteristics and physics to earth (though I am not sure how different temperate zones would come about) I agree that with others that life would develop separately in different locations. With infinity there is the possibility that completely separate forms of life (e.g., non-DNA based) would come into existence. I think the question is also interesting starting with a planar non-infinite landscape.

But in general, I think life forms would spread gradually over the landscape as they do on earth (though probably from different pockets of life on the infinite landscape). I do not see why, as one suggested, that we would continually (as a whole) be nomadic just because the world was infinitely flat. The answerer's premise was that we would run out of resources in one area and because of the infiniteness of the world, land and resources would not be valued and we would pull up stakes and move on. But just as happens here on earth (according to the premise), the land would replenish itself, and I think people would spread out naturally as population/resource pressures arose. The whole population would not pick up and move as nomads. And unless there was a scientific way to determine it, I do not think the population would realize the world was infinite so population migration would probably proceed much like it did no earth. They probably just wouldn't do the type of exploration we did to find shorter routes to places based on the curvature of the world.