There is the strong theory that the Earth and the Moon were part of the same core at one point. This would lead one to believe that there are precious minerals under the surface of the Moon.
Actually, the theory isn't quite like that. The leading theory for the formation of the Moon is of two proto-planets colliding to form the Earth and Moon system. The collision left the main planet (Earth) in-tact, but a large volume of debris in orbit. After the collision, the heavy material that was ejected sank back down to the primary body (Earth), while the lighter debris material stayed in orbit and went on to coalesce and form the Moon.
The result is that the Moon has a much lower density than Earth, and a much lower concentration of heavy elements. In addition, the lack of vulcanism on the Moon means that those heavy elements that did end up forming it are mainly deep in the core, making them inaccessible to mining.
Metals are normally heavier than dust and we don't want to send the moon out of orbit, so when we mine the moon we intend to put back as much mass as we take.
Seriously, the Moon isn't going to worry about a bit of mining -- it may be smaller than the Earth, but it's still a very very large body.
The Moon's mass is 73 quintillion metric tonnes (ref). You could take a 100 billion tonnes off it, and you'd barely cause a rounding error in that figure.
More significantly, if you're thinking of replacing the material that you extract... what exactly are you going to replace it with? Virtually any Earth rock you could think of (even the 'worthless' ones) would have more mineral value than the Moon rocks you're mining.
Will this prevent the Moon from going out of orbit?
What other astronomical challenges would we face?
No it won't.
Your biggest astronomical challenge will be if you think you're going to be able to transport all that mined material off the surface of the moon.
You would certainly be looking at an astronomical amount of rocket fuel!
You would be much better off leaving the mined material on the Moon and building things with it in-situ.