If the Earth's core was removed (gradually or instantaneously), the primary effect would be loss of the global magnetic field.
The mean distance of the Earth from the Sun would change by less than 1 part in a million.
Although the first two are factual, I think the surface would not sink significantly, at least in the short term.
The Earth's global magnetic field is due to the rotation of the Earth and the (slightly out of phase) rotation of the liquid core. If the core was removed, or just vitrified, the global magnetic field would collapse. This results in two issues that would eventually destroy human civilization (at least as we know it now).
The first issue is, without a magnetic field, the Earth would have no defense against solar storms (and less protection from cosmic rays). We haven't experienced a major solar storm for over 150 years, although one just missed us in July 2012. The 2012 event was about the same magnitude as the solar storm of 1859. One study estimated that, if a storm of that strength hit Earth now, it would take between 4 and 10 years for power and other services to be fully restored. And that was if we had a magnetic field!
The second issue is the erosion of the atmosphere by the solar wind. We have an example of that with Mars. Since Mars is smaller than Earth, it has a smaller heat capacity and a greater surface to volume ratio, so its core cooled much sooner than Earth's will. According to data collected from the Mars Global Surveyor by the late Mario Acuna, Mars' core began to vitrify about 1 billion years after its formation. That explains why, long ago, Mars had a sufficiently dense atmosphere to support liquid water on the surface, and now has almost none (about 0.007% of Earth's atmosphere).
Looking at Wikipedia, or any physics text, you will see that in the equation for the mean orbital distance in a two-body problem, the sum of the masses is in the denominator. The mass of the Sun is about 333,000 times greater than that of Earth. Since the core is a small fraction of Earth's mass, the change in the sum of the two, and thus the change in Earth's mean distance from the Sun, would be less than 1 part in a million.
The Earth's construction is complex and the exact mechanical properties of many of its components are not well known, so I can only say I think nothing catastrophic would happen in the short term.
It's the loss of the magnetic field that's the big deal.
/// Note: the description of the effects of loss of the magnetic field described above is a simplification of a complex event. Although it's true that Earth's magnetic field protects us from being directly bombarded by Coronal Mass Ejections (CMEs), it's the presence of a magnetic field that causes the greatest threat to our digital world. When bombarded by CMEs, the huge movement of the magnetic fields as they collapse induces equally huge currents in anything that conducts electricity. The result is almost all electric devices are fried--permanently. ///
In the links below, I've favored Wikipedia, as it is free for everyone, which is often not the case for scientific papers.
The Solar Storm of 1859:
The Solar Storm of 2012:
Calculating the Mean Orbital Distance:
Mars Global Surveyor Magnetic Field Mission:
How Earth's Magnetic Field is Generated: