What is the relationship between tyre pressure and weight? To give an example of what I'm trying to understand:

To accelerate to a given speed or maintain that speed with a tyre pressure of e.g. 100PSI requires a certain amount of energy. If that amount of energy is held constant, and the tyre pressure is dropped to e.g. 80PSI, how much lighter would my bike need to be to attain / maintain that speed.

What prompts my question is the observation that many cyclists who commute have under-inflated tires, yet have paid a premium for aluminium over steel.
It's not about energy to accelerate the bike to speed, but the energy to keep it there. For bicycles the kinetic energy is a small part of the total power output of the rider (typically under 10m/s = 36km/hour and 100kg, and e=1/2 m v² = .5 * 100 * 10² = 5000J or watt-seconds. So a casual rider putting out 250W could reach 36kph in 20 seconds, assuming no friction)

The relevant factor is rolling resistance, not weight. On a flat surface there's a more or less parabolic relationship between tyre pressure and rolling resistance. The region you're talking about is the low side of the curve, where more pressure in the tyre gives lower rolling resistance. The losses here are mostly from deforming the tyre, so riding underinflated is costing you money in increased tyre wear as well as time through riding slowly.

There's also a relationship between total weight and optimum tyre pressure. Vittoria have an online calculator and there's a simpler one here with a link to the papers it's based on. Basically, more weight means more pressure and it's not a linear relationship.

To answer your question, then: assume we're in a linear part of the curve just for simplicity{1}, with 80/100 of the pressure you need 80/100 the total weight. Assuming an 90kg rider with a 10kg bike = 100kg total at 100psi, that means you'd need a -10kg bike to keep the same rolling resistance at 80psi (90kg rider -10kg bike = 80kg total). Or alternatively, starting with a 90kg+10kg at 80psi bike, you could carry an extra 25kg at 100psi and have the same rolling resistance (100kg * 100 / 80 = 125kg).

From experience though, increased tyre wear is not a big factor. At one stage I was only riding about 8-9km each way to work so I dropped the tyre pressure on my slick tyre MTB to compensate. Tyre lifetime was still determined by tread wear and didn't seem to drop - it definitely wasn't halved, for example. Bike handling was compromised a bit and I did get more punctures (n~3, though)

{1} this is obviously not true, but it makes the maths possible. The real curve is very complex and specific to a particular tyre. It's also affected by road surface and riding style, among other things.