The atmosphere and water would be sucked into to the gravity well. You'd end up with no air on top of the platter, and the entire atmosphere in a tight sphere around the dense object.
To prevent this you could use walls around the edge. They would have to be high enough to prevent the air from pouring over the top.
Humans looking out would see right to the edge, and then a huge, impossible wall.
A small dense object (like a chunk of neutronium, or a black hole for example) will also generate a significant gravity gradient which would likely be noticeable to anyone living on the surface.
Gravity would be much stronger as you moved closer to the object, possibly lethally strong at the pole, depending on the thickness of the platter.
Gravity would always pull towards the object. Humans near the rim would have thinner air. Walking towards the edge would be like walking uphill. You would weigh less at the rim than at the pole, and would be able to jump higher.
Tendency to a sphere
Objects in space with sufficient mass will tend to become spherical. Rock is brittle. If it becomes unstable it will crack. You will need to account for this.
A note on stability
Ordinary matter would not be dense enough to have sufficient mass to replace the mass of the planet. You would be looking at some type of exotic matter, such as a singularity or neutron star.
At sub-planetary mass, neither of these would be stable, so you'd need some kind of technology to stop them evaporating in a wave of killer radiation. You'd also need some way to prevent the platter from collapsing into the gravity well.
The alternative would be to use some type of alien gravity generator.
A further note on solar radiation shielding
The magnetic core of the earth shields us from high energy solar rays which can strip away a planetary atmosphere. You will need some kind of solar shielding for long term survival. Perhaps your chunk generates a sufficiently powerful magnetic field.