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.
Most of the answers until now have focused on biology/evolution, even though you asked about cultural implications. One of them would be that all reference points would be, of necessity, local.

Each place on the infinite Earth-plane could and would probably be considered as the center of the universe (figuratively and/or literally) by its inhabitants. You haven't specified how this flat Earth would be lit, but a single sun (or light source of any kind) could never be enough, so there must be either some kind of luminous sky or multiple suns revolving "up there", for example. In any case there would be no north, south, east or west. Navigation would have to be done using notable local landmarks such as mountain peaks. In a flat plane the "horizon" would not exist; things would just appear to become infinitesimally small in the distance, in a clear day, or obscured by air itself (no mixture of gases is perfectly transparent), so people who happened to live in a place with no great mountains around would effectively lack all navigational clues.

Religion and science would include these things in their paradigms. They would have to cope with infinity. Some could imagine that reality must be finite and therefore everything outside a given radius from the local origin is an illusion. Legends could be common about some adventurer or prophet who travelled to the end of the real world and came back.