I would test this but I have no tools to test it with.
If I have a beaker of water and weigh it on a measure. Then I stick my fingers in the water (it doesn't over flow) does the weight change or does it pretty much stay the same (excluding fluctuations)
If I were to cut off my fingers then, and let them sink and hit the bottom would the weight change?
If I dropped something in that floated would the weight change?
Lastly, if I were to measure the water in a room filled with air. With a cieling 1 mile tall (atmospheric pressure = 1) and then measure the same water beaker in a room filled with air with a cieling 1 foot tall (atmospheric pressure = 1) would the measured weight change? Not the weight of the water but the weight of anything effecting the scale, including the air in the room, if that pushes down on the scale like I assume it does, I may be wrong