There's a lot of assumption that must be made here for this to work...I'm not entirely sure if life would ever evolve past single celled organisms as space scarcity is non-existent...it'd just expand forever in every direction as a one celled organism. It's possible that mammalian life would never come to dominate the planet as there isn't much room for mass extinction events either. If Humans did come to be, it would stand to reason there would still be dino's running around on the far frontiers...they did get several hundred million years to expand in every direction possible after all.
I highly doubt a working plate tectonic system could exist, which makes the land much more flat and I don't think you'd get much for mountains or deep oceans. I'd have no clue how geological processes would work in something like this either...infinite space = no pressure = no volcanoes? I'm not even sure you'd see hills.
No clue how suns and seasons would work...if at all. On a flat infinite planet, the suns rays would go off in all directions and you could probably see the next sun coming along way way off on the horizon (would there even be a horizon?). I doubt you could get a day/night cycle that even came close to resembling earth.
I'd suspect you'd get a series of lakes before you'd get oceans...tides wouldn't exist either.
Climate would be...well if air currents are influenced by the spin of the globe and a flat land has no spin, I'm really not sure if there would ever be dominant wind patterns. Rain is also affected here...if there were no strong trade winds to relocate rain to land, is farming even feasible? While I'm at it...it took millions upon millions of years for earths atmosphere to take the composition it has today...if there was a strong wind from one direction or the other, what happens when the air it draws is from a region that plant life hasn't made it to yet...would the air even been usable to us or animal life? A near infinitely large forest fire could create a cloud of smoke large enough to extinguish life as it floats over the infinite land.
Thermohaline circulation depends on a round globe (Thermohaline circulation is the 'great oceanic heat conveyor belt')...so heat distribution wouldn't be earth like. In Earths case, this would result in the northern atlantic, most notably England, Scotland, and Ireland to be completely frozen over. You'd most likely have extended glacial regions that reach from the area's that receive 0 sunlight and a middle band where the sun (suns?) travel on that are much warmer. Edit point - a flat land would be struck by sunlight evenly across it's entirety no? There might not be an arctic.
Would the planet have a magnetic field? Of course it couldn't exist naturally (just like the gravity situation) and would need to be handwaved into place...are there multiple magnetic norths? Is magnetism ever a possible method of navigation? I guess the stars wouldn't really move in this setup, so I would guess that stars would work for navigation quite well. Err..or maybe...you couldn't really have a moon as it'd eventually impact somewhere across the infinite. The stars really couldn't exist as they'd eventually come crashing down as well, no? With no magnetism and no starscape, navigation would be exceedingly difficult.
Is there a north/south pole, or is it infinite in these directions as well? I guess you'd need multiple suns to be over the planet at all times...actually you'd need an infinite number of suns over this infinite land wouldn't you?
Now to the cultural implications of a world of infinite size...I hate to say it, but you're in the range of 95 - 99% of this infinite plane existing in the realms of magic or if you'd prefer a 'god' crafting it...that same magic can be used to justify whatever cultural implications you feel like.