A strange but good question....
Here's a rather silly one which I still could not give a good answer to!
Say a large aircraft is flying along, full of many, many pigeons (enough to make significant weight) and they are all happily perched on whatever inside the cabin.
Now, the leader of this flock of pigeons (or parrots, magpies, ducks, toucans - whatever you want them to be) says "alright, lets all go flying around the cabin" and they do.
Does the aircraft's weight decrease as they all fly off their perches? Will it therefore climb (all other things being equal)? What if the aircraft is pressurised?
This sparked a rather interesting conversation at work!