What force does a floating object exert on the liquid it is floating on?

The floating object exerts exactly the same forces on the fluid as the fluid it displaced would have exerted. This is the essence of Archimedes principle: the rest of the fluid does not notice that the displaced fluid is gone.

My answer is going to be a bit different from others but give it a try.

Newton's third law says that if a body $A$ exerts a force on a body $B$ then the body $B$ will alao exert the same type of force of same magnitude and in opposite direction. The origins of the both the forces are same.

What do you think why are all water molecules in a container not falling down due to gravity (considering in vacuum) ?

This is because every water molecule is pushing or repelling every other water molecule since the distance between them is less due to gravity. See the graph of variation of intermolecular forces with separation between two atoms

When you drop a body in it, the water molecules are pressed more against each other and the repelling forces increase (notice the change in repelling force with a little change in distance). Also there is a repelling force between the atoms of the body and the water molecules (for the same reasons), which contribute in its floating.

You can see that the force between two molecules are acting on both the molecules and the force is of the same type. So, the floating body also applies the same type of force on water molecules.

Apart from this , increased force between water molecules also contribute in rising of water level (if you want to know how , comment below).

Hope it helps ?.