Why does weight increase if an object is held in water? If you put a glass of water on a scale and partially submerge an object in it while still holding the object in hand, the scale readout shows an increase in mass. It happens if the density of the submerged object is lower than the density of the water (you can feel the force you need to apply), but also if the density of the submerged object is greater than the density of water.
Why does the weight increase for denser then water objects even if they are still being held?

 A: When you submerge an object of certain weight $w$ in a fluid (water tank), there is a buoyant force exerted by the fluid to the object $B$ pointing upwards. This means that the upward force on the object exerted by the hand $F$ is smaller for the magnitude of the buoyant force
$$F + B - w = 0 \qquad \rightarrow \qquad F = w - B$$
Third Newton's law of motion teaches us that for every action there is a reaction. That means that if the fluid is pushing the object up with a (buoyant) force of magnitude $B$, the object is also pushing the fluid down with a force of same magnitude. Since the water tank does not move, it follows that the (ground) surface has the push the water tank up with a force
$$N - w' - B = 0 \qquad \rightarrow \qquad N = w' + B$$
where $w'$ is weight of the water tank and $N$ is normal force exerted by the surface. It is exactly magnitude of this normal force that the scale measures.

Why does the weight increase for denser then water objects even if they are still being held?

This has something to do with how buoyancy works. By definition, the buoyant force magnitude equals the weight of the fluid which has the same volume as submerged part of the object.
If object is less dense than the fluid, it will float on the fluid and you do not have to hold it. When object floats on the fluid without you holding it, then $F = 0$ and $N = w + w'$.
If object is more dense than the fluid, it tends to sink and you have to hold it if you do not want the object to sink. In that case the scale would measure $N = w + w' - F$, and if you decide to let go of the object then $F = 0$ and $N = w + w'$.
A: Many hydrostatic problems can be understood if we remember that static water does not transport any information beyond the pressure it exerts. The walls and bottom of any enclosures, like this glass, do not know where the pressure comes from, and they don't care. All they do is experience the force from the pressure.1
When the water level rises, the water pressure acting on the bottom of the glass and the horizontal "portion" of the walls grows correspondingly, exerting a larger downward force which the scale measures.
Whether the water level rises because we add water or because we immerse a piece of styrofoam or because we immerse some platinum the glass cannot know.

1 One example for this principle is an imagined Hoover Dam that just dams a one meter thick, 400m wide, 200m high slab of water, instead of Lake Mead, with an imagined mirror copy on the other side. The dam would need to be exactly as strong for one meter of water as it needs to be for the entire Lake Mead. The dam doesn't know anything beyond the molecules touching it. Nor does your glass.
A: The object requires some total amount of force to be at rest. When you partially submerge it, it will definitely experience a buoyant force, and it will feel slightly lighter to you, due to the buoyant force. That means that something else must be holding the rest of the weight, and that must be the buoyant force, and ultimately the scale.
$mg = F_{scale} + F_{you} = F_{buoyant} + F_{you}$
