It's worth noting that there's already a technical term for 'hole through a planet's crust into the liquid regions': Volcano. The most likely 'realistic' answer to your question is 'they create a new volcano, it buries their equipment in lava, and they all go home sadder, wiser, and empty-handed'.
It's also going to tricky to pump the liquid to the surface - the pressure's not going to be enough to push iron/nickel that far using suction. You can actively pump it using positve pressure - but that means building a pump that can run while submerged in 4300K high-pressure molten iron. That's going to be... difficult.
But while 'mad scientists create a new volcano and lose control of it' has some definite potential as an adventure setting, I don't think it's quite what you were looking for, so I'll take off my hard-science cap for a bit and break out my emergency unobtanium supplies.
Drilling Corp Ltd's top Russian engineers have developed new, Russian nano-tubes, which are far stronger than the effete Western ones that everyone else was using, and their Western-trained legal department is confident they can keep the case of DCL v Reality tied up in appeals for the next decade; in the mean time, they plan to siphon away Dirt's* core and ship it home to sell at a literally astronomical profit.
The initial drilling process is probably going to be pretty uneventful, on a global scale. It's going to take some very heavy-duty reinforcement to make the process work at all, but once you get past that, it's pretty much just digging a hole. There's not necessarily going to be significant pollution - a lot of the nasty stuff involved in real-world mining is from refining the ore, not the digging itself. (Of course, you can attach as many side effects as you want - there's just no reason to assume they're automatic.)
Long-term, the pumping process is definitely going to change the planet noticably - but it might be very, very long-term. The core of an earth-like planet is very, very big: the liquid outer core has a volume of roughly 1.5 x 1020 m3 - and that's assuming you could keep it at the same pressure as the core itself. If you can't (and you almost certainly won't be able to), every drop of liquid you extract is going to expand even further as the pressure drops. For reference, that volume is:
- Enough to fill around 470,000,000,000,000 of the largest supertankers ever built.
- Roughly as much as 9,000 modern deep-sea oil wells would pump if you ran them for as long as the Sun has existed to date (4.6 billion years), assuming that they ran at maximum rate the whole time.
- So much that if they drill a 1km radius pipe, and pump out the liquid at 100 miles per hour, it would take almost 34,000 years to finish.
No matter how you look at it, this isn't going to be a quick job.
But Drilling Corp Ltd. doesn't care about that, and they're forging full steam ahead. What happens to the planet?
The structure of a planet is far more malleable than you might assume - we're not going to leave any holes in the center, or anything like that. Gravity will pull the rock above the core down, and keep the planet as a solid sphere. From the perspective of the inhabitants on the surface, this means a lot more earthquakes and volcanoes as the crust reshapes itself to match the new, smaller size. It's going to be unpleasant, definitely - but see the above points about this being a very long process. Unless DCL is pumping material out at an ungodly rate, the effects on the surface are likely to be closer to the 'highly inconvenient' part of the spectrum than the 'end of the world' part. The really big changes - seas rising, continents sinking, etc. - will happen slowly enough to see them coming and evacuate. People will move away from cities, and start making buildings that can handle the earthquakes - low, strong and flexible. Coastal areas will be unpopular - too many tidal waves - as will areas near fault lines and other geologically active zones, where the majority of the earthquakes and volcanoes will hit.
Running some numbers, we've extracted a bit over 14% of the planet's volume, which means the radius is going to drop by around 5% - close to 320km straight down. We've also removed around 27% of the planet's mass, so gravity is going to go down to 73% of Earth-normal. The magnetic field is gone, since that was produced by the movement of the core.
Otherwise, the planet looks surprisingly normal. The maps have been rearranged significantly - oceans shifted, mountains risen, continents sunk into the ocean - but the surface climate is driven by the sun, not the core, so that hasn't changed much. Life will still live there, of various kinds. Some (perhaps many) species will be extinct, but the remaining ones will be tenacious, quick to adapt, and very hard to kill - and will almost certainly include humans.
And now that their planet has settled down again, they're free to turn those abilities to something other than mere survival. Like, say, seeking revenge on the people who wrecked their planet in the first place...
*The third planet in orbit around The Star, located in the Creamy Way galaxy.