Q:

Hitting the tipping point for core removal to degrade atmosphere retention on a moon

Let us assume that we have a moon with a reasonable iron core enough that it can retain atmosphere, (as per this question), now let us further assume that the people who settle on the moon are short sighted and start mining deep into the moon while also building up a breathable atmosphere. The ores are then processed and shipped off-moon to the planet.
At some point the moon is going to hit an environmental disaster wherein the mass is reduced such that atmosphere starts to leak.

How quickly would this become noticeable (assuming no one is really looking for it). Would it be gradual or sudden?
Would the planet capture all or most of the lost gasses?
What other effects are they likely to see? Would the orbit change, for example?
How much material would need to have been removed for such a "disaster" to start?

A:

You are massively underestimating the scale of planetary bodies.
World production of iron is roughly 2 billion metric tons per year.  Let's be crazy and say that mining of the moon is 2 billion metric tons per second.
In that scenario it would take over 11,000 years for the moon's mass to be reduced by just 1%.
So realistically, the answer to your question is never.

A:

Not only is Dan Smolinske correct (well, 100,000 years to lose 10% of the mass is probably as close to never as is reasonable to think of), atmospheric loss will not become catastrophic. As the moon's mass is reduced, the atmosphere will expand, and "sea-level" atmospheric pressure will be reduced proportionately. For a constant diameter of the moon, since all the mining is of the core, a 10% reduction in total mass (100,000 years, remember) will cause a 15 psi atmosphere to drop to 13.5 psi at sea level. That's about the pressure at 2700 feet, and is hardly life-threatening.
I'd worry more about the structure of the moon being weakened, resulting in a slumping of the entire surface along with truly horrendous moonquakes.

