Does it take significantly more fuel to fly a heavier airplane?

In your own question you recognize that the Bernoulli equation is the wrong thing to apply to this situation, because obviously there are dissipative losses involved.

My preferred way of looking at this is recognizing there is a lift to drag ratio that exists as a metric for aircraft. This can be 4:1 or 25:1 depending on the plane. Regardless, provided that we accept the existence of this ratio in the first place, then the airlines are justified in the claim that more weight $\rightarrow$ more fuel. Limiting the discussion to cruising, it then becomes a simple multiplication of weight times lift to drag ratio to find fuel use.

The other flaw in your argument is, of course, the assumption that speed can be increased to compensate for more weight. A cursory reading into the flow path of turbo-machinery will disprove this. The jet engines will be most efficient at the designed cruise speed and rotation speed, and any deviation from that will alter the angles at which the air hits the rows in the turbine, causing efficiency to decrease. In the real world, drag also tends to increase as some power of velocity, which in itself will probably predict some marked decrease in the lift to drag ratio, again, making the plane consume more fuel. If the plane uses different altitudes to compensate for different weights with the same velocity, then more dense air will obviously cause more drag. It's true that these are ultimately viscous losses, but this flow is turbulent, and its likely that drag will scale as something close to $\propto \rho v^2$ (density times velocity squared) as a result of that fact. As the density increases fuel consumption will too.

One thing in your argument is that more lift, means a higher speed. This may not be what airliners do. Airplanes (at long flights) choosse their cruising altitude based on their weight. Higher weight means lower altitude. I think this should be included in the incremental cost calculation of additional piece of luggage.

First, simple Google hit: http://www.ehow.com/about_4572148_why-do-planes-fly-feet.html . Some of the physics aspects are however mentioned here, that can be used in your derivation.

Lift is roughly proportional to angle of attack, and to speed squared. As a pilot, you instinctively balance these two.

ADDED: Like if you suddenly drop a heavy weight, making the plane lighter, its lift isn't any less, so it starts to accelerate upward (climb). You notice this and either push the nose down with the trim wheel (lessen the angle of attack, making the plane go faster at the same power) or reduce throttle to reduce speed because you need less lift at the original angle of attack. Or, you do both, and stay at the same speed.

Drag is the sum of parasitic drag (that's mainly your viscosity) and induced drag (drag due to lift). More lift, more induced drag. More drag, more power needed.