As the question asks, what would happen to the Earth if its core collapsed? Now I don't know if this is something that is possible or even realistic, but I found the question interesting nonetheless. I looked this up and the best I found was gravitational collapse, but that has more to do with the formation of stars than it does with planets.
The earth's core is a blob of molten metal and stone, already compressed as far it could be under its own gravity.
As far as we know, there are no hollows or bubbles inside the earth, because the total mass of the Earth matches with what we know of its composition and densities.
There is at least theoretical way for the core to collapse. Drop a small black hole into the earth.
If the black hole is too small, the collapse will not occur as the black hole will lose mass-energy via Hawking radiation faster than it gains from infall.
A slighter larger black hole, could take a long time for its effects to be felt, but all black holes larger than a certain threshold would eventually collapse the core.
Of course, the collapse would not stop with the core, as the block hole growth would accelerate and eventually consume the whole earth.
Needless to say, lots of bad things would happen during this process. But before the effects are noticeable on the surface, the effects could be measured by neutrino detectors, micro earthquakes and gravity anomalies.
This may be a bit of a tricksy answer, but the Earth's core probably is collapsing. Very very very slowly. For the definition of core that includes both the inner and outer core.
The inner core is solid. The outer core is liquid. It is believed that the inner core is denser than the outer core, and that the outer core is slowly crystallising onto the inner core. Or at least, changing from a liquud to a solid phase. The core, overall, is therefore becoming more dense and is shrinking. It is very slow because it can only happen as fast as the heat released by the phase change can escape. It is a long way to the surface!
The "probably" relates to uncertainty over the exact composition of the cores and the behaviour of what is mostly nickel-iron at the incredible pressure and temperature at which this solidification is happening. It is completely out of range of lab experiments, and we cannot obtain core samples.
Effect of this "collapse" over human timescales at the planet's surface: none. A billion years from now, the planet will have become slightly smaller.