Mining on the Moon, and by that I mean removing mass from the Moon, I believe will affect the equilibrium of the planet and cannot be compared to mining on Earth. When we mine on Earth we don,t remove material from the planet we only chemically or physically alter it. Who can predict what effect even the slightest man made changes to the Moon will have over future millennia?
Mining on the moon would remove some mass, but I don't think you'd notice.
First mining on the moon probably won't involve strip mining and hauling all of the material away. Instead rock will be dug out, the ore poor rock will be discarded back on the moon as slag (no mass loss), and the ore will be processed (slight mass loss).
Mined material would likely be smelted on the moon which means more slag being left behind. The amount of actual material that would be removed is so small that it won't have any affect on the moon or on Earth.
To give a little perspective, here is a Reddit post that asks the opposite question: How much mass could we add before it would affect the moons orbit
And since it's not long, here is the post text in case it disappears in the future:
If humans colonize the moon, at what point would the amount of mass we add to the moon (buildings, people, etc.) be enough to significantly change it's orbit around earth?
I'm going to do some calculations here aiming for a .001% change, which I don't think would make a whole lot of difference in terms of effect on Earth, but should illustrate the scale of the issue.
The mass of the Moon is 7.35 x 1022 kg. That's...that's such a huge number it's crazy. If we wanted to make a difference of a thousandth of a percent, we'd have to put 7,350,000,000,000,000 people on the Moon (average of 100kg each). Let's make that a little more accessible, since 7.35 quadrillion isn't a very easy number to wrap your head around (thanks to those who pointed out that I got the label wrong. Even after I thought I got it right again).
The Empire State Building weighs approximately 350,000 tons. That's 317,514,659 kg. If we wanted to change the mass of the Moon by 0.001%, it would take 2,314,853,752 Empire State Buildings added to the Moon. In other words, you'd have to build an entire Empire State Building for every 3 men, women and children currently on Earth to make even that small change.
And then there's the question - why would we add that much mass to the Moon instead of using the materials already there to build what we need? It's extremely inefficient to take anything on Earth and put it on the Moon, so we'll probably just move people and a bit of initial material, and then build what we need with the rock and other materials on the Moon.
And I think we're safe...the number of people above? That's enough people to cover the entire moon with over 190 people per square meter of Moon surface, so it'd be pretty crowded.
No appreciable effect can be achieved by mining.
The mass of the Moon is 7.347 × 10 22 kg. Sensibly alter that amount is way out of our technological ability. Even removing 1000 kg of Silicon per second it would take 10 19 seconds to consume it, which is 100 times the age of the Universe.
P.S. Today seems to be the mass order of magnitude day...
Just gonna throw some figures here.
The world mined around 2.28 x 1012 kg of iron in 2015.
The mass of the Moon is around 7.35 x 1022 kg (eyeballing and rounding up here).
So if we were to mine the moon for iron only, with all the countries of the world combining their efforts and taking from the Moon's surface at the same rate they can mine today on Earth, and sending that mass somewhere else.
It would take in the order of 106 years (a million years) to reduce the mass of the Moon by 0.01%.
I think we would be safe for a long while, at least where the stability of the Earth-Moon system is regarded.
The mass of all solar system objects continually changes. They all accumulate mass from cosmic dust, meteoroids, asteroids and comets.
It is estimated the Earth accumulates between 30 and 180 tons (27-163 metric tons) of dust each day, which is between 10,950 tons and 65,700 tons (9933 - 59,600 metric tons) annually. On the losses side of the equation, Earth loses around 50,000 metric tonnes from its atmosphere annually.
Similarly, the Moon will accumulate mass but lose much less than Earth does, due to its almost non-existent atmosphere.
The structure of the Moon is such that it has mass concentrations, known as mascons. These affect the gravitational field of the Moon so that most orbiting space craft will not maintain a long term stable orbit around the Moon, without corrections.
The construction and filling of the Three Gorges Dam in China created an artificial mascon on Earth. This has affected Earth's rotational speed.
As on Earth, large scale surface mining on the Moon could create large open pits and large waste dumps. This would be a localized redistribution of surface material. It could change the nature of the mascons, but it would only be a very minute change. They are unlikely to change the rotation of the Moon.
Any mining on the Moon may redistribute a lot of surface material, but any the amount of metal removed from the Moon would be very small due to concentration and smelting practices.
As others have stated here, the affect of mining the Moon will have an exceedingly small effect on the Moon.
Don't forget, that you first need to take material to the Moon (mining equipment, etc.). Then, as transport costs between the Moon and Earth are quite high, you would want to do as much extraction as possible on the Moon.
Think about hauling ore with 1% grade vs. concentrate with 95 % grade. So, even if mining moves megatons or gigatons of rocks per year, the actual tonnage of material carried away is orders of magnitude smaller, provided you do mineral processing and extraction on site.
Note, that such procedures are followed on Earth today, with its non-astronimical freight costs (pun intended).