I’m not sure what “flat earthers” actually believe, but assuming the world is literally flat leads to some interesting results. For the sake of keeping things simple, lets assume that the world is flat like a coin: locally bumpy, but overall shaped like a disk. This assumption is based on what the word “flat” means. A saddle isn’t flat, so a hyperbolic-curved earth doesn’t count as “flat” in any reasonable way. Further, lets assume that it keeps the same surface area that the shall we say “conventional” model claims that it has. This assumption lets us avoid the absurdities  required to preserve distances, and so travel times, while unwrapping a sphere to cover a disk.
These assumptions tell us how big the disk is. The earth’s surface area is allegedly 196.9 million square miles. A disk’s surface area is given by pi * r2, so do the math to get r and you wind up with a radius of 7.916 million miles. This may present astronavigation problems, as the moon is only 238,900 miles away, so it might hit the disk… if the moon were real! 
Ok, so we all live on a disk that’s about 16 million miles across. Because I’m comically egotistical, I’m going to say that my home town is in the middle of the disk, equidistant from every edge.
Now we come to an interesting point: How thick is the disk? Let’s assume that we believe in gravity. The gravity in my home town is one G, and it causes objects to fall down, which is to say “towards the surface of the earth”. Under the conventional model, this is because the earth is under me, and so the gravity caused by the large mass of the earth exerts a force on objects above it. Now there are two ways we can go: either the earth is made of the stuff that it is observed to be made of, and has the density it is generally observed to have, or it’s made out of something far more dense. All that this really varies is how thick the disk is under my home town. To have 1G there, using the conventional materials, requires at least the alleged thickness of the round earth, which is to say about 7,917 miles. Using something denser makes it thinner, without affecting the gravity, but there are limits on how dense matter can get.
Ok, so far, so good. However, we’ve set a little bit of a trap for ourselves here. Everywhere you go on the surface of the earth, the gravity is about 1G, so everywhere you go, the disk earth has to be about 8k miles thick. In my home town, at the center of the disk, this isn’t a problem, because the gravity in all the other directions balances out.
What’s that you say? Yes, gravity in other directions. You see, we’re talking about a disk that measures about 8k miles thick and 16M miles across. If you’re in the center, there are equal amounts of disk around you in all directions, so the pull in all the other directions balances out. If you’re off center, there’s more mass on one side of you than on the other, and so there is a component to the gravitational pull that isn’t straight down, and is unbalanced by the ratio of how much mass is on each side.
Initially, this would probably be pretty subtle. Things would fall a little bit in the direction of my home town, but more or less straight down. Friction would suffice to keep things on surfaces, but round things would always roll towards my home town. It would only be a little harder to walk away from my home town than towards it. I doubt it would affect which way water goes down the toilet all that badly, although the water would pile up on one side of the bowl. However, as you got towards the edge, things would get FUCKING DIRE.
How dire? Well, lets look at the volume of that disk earth, which is properly a cylinder now that we know how thick it is. A cylinder that’s 16,000,000 miles across and 8,000 miles thick has a volume of 6,430,000,000,000,000,000 cubic miles. The conventional model earth has a volume of 260,000,000,000 (that is to say, 260 billion) cubic miles, and exerts a gravitational pull of 1G when you’re on the surface, which is to say that all of it is under you. When you’re on the edge of the disk earth, which is to say that almost all of it is, say, east of you, it exerts a gravitational pull of (around) 24,730,769.2308G. So to understand the force exerted on some poor schmuck who happens to get teleported from my town to the literal eastern edge of the world, assume that the ISO standard schmuck weighs 150 lbs. He will suddenly experience a thrust of about four billion pounds of force to the west. For comparison, a Saturn V rocket generated about 7.5 million (with an “m”) pounds of thrust, or about 500 times less thrust than people living on the edge experience as a consequence of just being there.
Unfortunately, since some of that force (at least 1G of it, thanks to the thickness of the disk) is downwards, towards the center of mass of the disk, rather than the location of my home town, the schmuck is going to hit the ground going absurdly fast and get spread all over it.
But wait, what is that ground made of? We said earlier that this disk is made out of normal earth stuff, which you can go out and observe to be mostly silicate-based rocks. The rocks under my home town have about 8 million miles of rock around them, pressing in towards the center of mass of the disk. Extremely weird stuff happens to matter under those kinds of pressures. Hydrogen (theoretically) becomes a metal. Atomic nuclei get mashed into each other.
That said, my home town is going to have other problems. For example, all the water in the world, and all the air, and all the stuff that’s far enough away to experience mostly-sideways gravity, is all going to flow towards the center of mass of the disk, and some of it will be coming in very fast. Since my town is barely above sea level under the conventional model, I think it’s going to get both extremely hot, due to the abrupt change in pressure, and rather wet, although possibly not before the disaster of degenerate matter that’s forming under it gets to the surface.
Alright flat-earthers, you got my home town into this, you get it out. Why are none of these effects observed on the ostensibly flat earth that we live on?
Well, maybe it’s not 8,000 miles thick. Maybe, it is in fact quite thin, and They can manipulate gravity to provide ~1G everywhere you go. They put some thickness everywhere that anyone decides to dig, or anywhere a tree falls over, but everywhere else it’s…. 1 inch thick. One inch is pretty thin, but that still puts the volume of the disk around 99,804,672,000,000,000 cubic miles, and so the inwards G-force experienced by someone on the edge at around 383,864.123G. This is still a troubling amount of force, but clearly if They can provide 1G over the whole surface of the earth, They can sort this out too.
That said, how long have They been doing this? If the earth has always been a disk, then obviously They were doing it before we evolved, so they’re not human. If They did it recently, why did no one notice the change? Was that the “road work” that was making my commute to work slow this morning? Either way, this moves “Them” from the category of “Federal/NWO government conspiracy to hide the truth, man!” to “Capricious god or gods with odd senses of humor”. At that point, there’s no use arguing what shape the world is, because it might be different tomorrow.
 These absurdities extend from the simple “everyone who has ever traveled is part of the conspiracy and lies about how long it takes” to the complex “THEY (It’s always ‘they’, innit?) can alter spacetime to slow or speed up travel”.
 Spoiler alert: it is.