Disco,
I think the aircraft being airborne is just a distraction getting in the way of a good story...
If you draw a control volume around the aircraft, the aircraft and everything in it could be represented as a single mass at the CofG, gravity applied to the mass gives the weight.
If they all start flying, the birds are still in the control volume, the total mass, hence weight would not change (no fuel/oil burn change assumption made).
The only change would be a very very minor change as the CofG which would move very so slightly away from the earths CofG, and gravity reduces in proportion to 1/r^2, where r is the distance from earths CofG, so yes the weight will reduce, but very very very small amount, and yes if airborne you will have excess lift so an upward acceleration of some very very very small magnitude would be expected.
A pressurised aircraft would weigh more than a non-pressurised aircraft, just due to the higher mass of air inside, i.e. a fully oxy cylinder weighs more than an empty one, higher mass of oxy in it.
Edit : should have put an upwards had a upwards