If there were an absolutely terrible accident, and part of Earth's core was displaced toward the surface, making a bubble of liquid nickel and iron, what would be the major problems with this?
You ask about a portion of the core being displaced toward the surface. This would move the center of mass toward the surface of the earth.
How much of the core moves, what the shape is of the part that moves, and how far that bit of core moves all matter when determining the effect. If, for instance, some of the core reaches the surface we have a new kind of volcanic activity that could make the Yellowstone Traps seem insignificant. For a less dramatic movement, the core being too close to the surface could result in much higher geothermal heating for that region, perhaps turning the Mediterranean Sea into a gigantic pot of bouillabaisse.
For this answer, I will assume that the core doesn't come to the surface, and that the mantle is essentially intact. Some part of the core moves "outward" from the core so that the core grows a bulge and is no longer best approximated as a sphere. Since the core is denser than the surrounding structures, the bulge moves the center of mass.
The center of mass is really important. It defines the point through which the planet's spin axis passes. The drama of the effect would depend on how much the center of mass were displaced. To avoid the "everybody dies" scenario, let's assume it moves a small amount, say about 7 miles. That can't be so bad on a 7000 mile diameter sphere. Right?
Except that it can. One change is that the water will flow so that it forms a smooth sphere around the center of mass. The low part will be covered with an added 7-mile deep ocean. The high part will be higher than the tallest mountains. If displaced in the right direction, the Mariana trench could become a salt lake above the new sea level as it is left to dry.
Another change is that the air will also recenter on the center of mass. Those former sea-level plains which are now 7 miles high will be uninhabitable without pressure suits.
The third substance that surrounds the core will also, slowly, flow to once again equilibrate to a sphere, and the earthquakes will be spectacular as the surface rock and mantle slowly flow and reform.
In summary, the low side is lost to deep water. The high side is lost to anoxia. The middle would still be habitable, provided you survive the earthquakes.
But, maybe 7 miles is more than you have in mind.
Even 500 feet would completely reform the coast lines. Dial in the displacement you want, look at topographic maps of the earth for the starting configuration and cover or expose whatever you'd like. On the low end, climate change for every spot on the planet. On the high end, a grand experiment in reshaping life after nearly all life on land and much of life on the ocean floor is destroyed.
The major problem is that the mantle is not made of liquid magma as Holywood likes to portray. It is rock solid, no pun intended. The fact that it moves on a geological time scale helps keeping the lay idea of a liquid mantle alive.
So, the core went to the surface. That presents three problems:
Moving it there. It won't be like a bubble working its way from the bottle of a cup of sparkly water. It will be like moving the seed of an avocado past its shell by pushing from the outside with a chisel and a hammer. What is on top of the core will be moved out of the way just like that.
Gravity. If the Earth was just a small asteroid, the kind that is just a handful kilometers wide, we could stop at the avocado analogy above. However, you are moving up to a third of the mass of the planet towards its surface. That's a lot of mass. The side distant from the core will be pulled towards and collapse on top of it. Then the whole planet will rearrange itself around the new position of the core and become round again. Remember the IAU's definition for what is a planet (emphasis mine):
a planet is a celestial body which (...) has sufficient mass to assume hydrostatic equilibrium (a nearly round shape)...
- Heat. The amount of energy involved here is much greater than things likes detonating all the atomic arsenal, or being hit by dino-killing asteroids multiple times per second. The heat generated by this will melt the planet. And I mean melt - the crust will become a proper liquid, unlike the mantle as it is right now. The oceans will vaporize and together with the atmosphere escape to space, and we probably won't have another bombardment of icy asteroids again, so the new Earth will not be able to harbor life (as we know it) after the Hadean this core displacement event would cause. The new Earth could look more like Venus after a couple billion years, but with more seismic activity.
The core is the densest part of our planet, being made, as you state, of iron and nickel. Offsetting it while the planet keeps rotating means changing the momentum of inertia of the whole planet.
This will induce quite some shaking on the entire planet, altering the day-night cycle and inducing quite some earthquakes while things settle down again. Think of what happens to the wheels of your car when they are not balanced.
Moreover it will also disturb the dynamo generating the magnetic field protecting us from the solar wind. And also the magnetic poles will be shifted.