Intuitively I would think that the more nectar you put into the reservoir the more weight it will put on the bottom piece (the tray that holds the open pool of nectar that the birds can feed out of). While I was trying to think about this I thought about doing some free body diagrams. The reservoir would have a force acting upwards on it from the hanger, and a downwards force from the sheer from the stored nectar. The bottom piece would have an upwards force from where it threads onto the reservoir and I can't really figure out how to do the downwards force. I'm thinking if I could calculate the pressure all along the interface between the pan and the nectar that would do it..
The force the fluid does on the bottom piece does not depend on the height of the water column of the reservoir. It does depend on the height $h$ of the water column in the plate.
This can be easily seen by the fact the water is static so the pressure at any horizontal plane is the same. The pressures in $a$ and $b$ are the same. The force of the water on the plate is $F=pA=\rho g h A$, where $\rho$ is the water density and $A$ is the area of the plate.
It is definitely counterintuitive so I had to go get an empty bottle of wine and a plate to check it. As long as the bottle does not touch the plate, the latter weights the same no matter if the bottle is full or almost empty.