Will a bathtub full of water weigh more if I add something that floats in the water? Let's say I have a bucket or something bigger, like a bathtub full of water. I weigh that bath and get some value, $x$.
Then I add a small boat made of wood to that bathtub that doesn't touch the borders (i.e., it is only touching the water). 
Will this boat add to the total weight of the bathtub, or does the weight of the tub stay the same?
 A: Mass is mass.  If you add something which has mass to the bucket, the bucket now has more mass.  It doesn't matter if it was more dense or less dense.  If you add 1kg to a bucket, you add 1kg.
Now there are two areas where this falls apart.  One is in the case where the object you are adding is actually less dense than the air (not just less dense than the water, but actually less dense than air, like a helium balloon).  In this case, we can forget about the minor detail of whether the object is floating on the water or not, we can focus on the entire bucket which is immersed in air.  This object displaces a larger mass of air than its own mass, so it will actually add lift to the bucket.  The mass of the bucket will still be bucket+water+object, but the bouyancy forces will make the bucket feel lighter.  Indeed, this is precisely the mechanism used by hot air balloons.
The other corner case is the case where the bucket is already filled to the brim with water.  Adding a single drop of water would cause a drop to have to spill over the brim.  Now when we add our toy boat the situation gets a little more complicated.  The boat will displace a mass of water equal to the mass of the boat.  This displacement will cause the water to rise and spillover.  If you tally up all of the masses in this case, you will find that the bucket+water-in-the-bucket+toy boat will have the same mass as the bucket+water did before you added the toy boat.  Where did the extra mass go?  There's a puddle on the ground outside of the bucket, whose mass is exactly equal to that of the toy boat.
A: You can answer that yourself with a Gedankenexperiment:
What if the boat you added were made of water? (BTW, that's not hard at all: an iceberg is just a boat made of water.)
A: Yes it will, the total weight will increase. 
From Archimedes' Principle 

Suppose a rock's weight is measured as 10 newtons when suspended by a string in a vacuum with gravity acting on it. Suppose that when the rock is lowered into water, it displaces water of weight 3 newtons. The force it then exerts on the string from which it hangs would be 10 newtons minus the 3 newtons of buoyant force: 10 − 3 = 7 newtons. Buoyancy reduces the apparent weight of objects that have sunk completely to the sea floor. It is generally easier to lift an object up through the water than it is to pull it out of the water.

The word apparent is important here, it is relative to the water, but the net weight of water and whatever you put into it, even it floats,  will increase.
A: Not only does adding something that floats cause the system to weight more, hanging something that doesn't float from a string and lowering it into the water without letting it touch the bottom or sides of the bucket causes the system to weight more.
Seriously. I do a demonstration of this in class on a regular basis. I put a beaker about 2/3 full of water on a beam balance and adjust the whole until it balances. Then I lower a brass or steel weight in on a string without letting it touch the sides or bottom of the beaker. The scale dips on the beaker side.
There are two ways to understand this:


*

*Newtonian reaction If the fluid pushes up in the object (buoyant force), then there is a 3rd Law reaction force on the fluid that is supported by the beaker and in turn by the balance.
Moreover, you can measure the buoyant force by adjusting the beam weight until it balances once again and subtracting this new weight from the old one.

*Static pressure When we lowered the weight it it displaced some liquid upward, making the column in the beaker deeper than it had been. The pressure of the water on the bottom consequently rises.
(In both cases I have assumed that the beaker is not full to start in the way that Hot Licks mentions in a comment.)
A: Yes, it will weigh more; you only need to consider the bucket-water-floater/sinker system as a whole to answer this.
Whether or not the object floats simply decides which forces keep the floater/sinker in the system. The floater/sinker thrusts downwards on the water, which thrusts upwards through the buoyant force. If the body sinks, it rests against the bottom of the bucket, which also pushes up on the body, so in that case both buoyant and bucket reaction forces support the body. 
A: The bucket is carrying both things: water and boat.
Otherwise I would be able to carry anything in a cup of thick enough liquid (if I could balance it). 
The fact that a buoyancy force holds up the boat is no different than when a tray holds up cups and plates with a normal force. Newton's third law says that such force gives an equal force downwards on whatever holds it up. You still feel the weight of both. 
A: Of course it will, consider it like if I put a weight that gets support from your head and weighs you , certainly your weight would increase.
OR
Considered a big baloon of air if I press it i.e. apply pressure on top of it.( equivalent to add a weight) and place it in a  weighing machine it would show increase in its mass.
A: Let's do a thought experiment: Put your bucket on a sensitive circular scale free to slide up and down to show the weight and rotate to accommodate the bucket.
when you lower your boat on the surface of the water the scale will read a spike a bit more then the total weight of the bucket and boat together because it is absorbing the kinetic energy of the down velocity of boat and breaking its fall.
Then the scale will start to go up and down in a dying sinusoidal vibration damping the impact of the collision with boat.
finally it will stop steady showing the total weight.
Other interesting things will happen, e.g. if lowering your boat you touch the bucket and make it swivel on the scale it will read less Wight because part of the total weight is now supported by the walls of bucket due to centripetal forces.
By placing your boat in certain points you could create nice looking standing waves!
If you had a very sensitive laser probe you could measure the lowering of small boat's effect on the orbit of the earth!
A: Yes weight will increase ,what machine measure is mg =where m is mass of bucket+mass of water +mass of block .that,s how newtonian mechanic ,s works ,when we think about the answer of this problem question comes in a mind is what about the force that push block upward  ,again newton law block also push it back.
