All sane engineers would walk away unless they are paid by the hour without penalty for failure and they have ethics of Wally because it is far beyond engineering and science that we know.
1) You mention removing the core -- this is even way more insane than drilling through the core.
2) How would you combat the temperature? No material can survive the temperatures required. The wall of the tunnel will melt or vaporize. You can't even cool the walls because part of the tunnel would necessarily be in contact with the core (and under extreme pressures - up to 3.6 million atmospheres) so even running liquid helium through the tunnels walls would not prevent its failure.
BTW, the tiniest leak would result in explosively filling the tunnel with magma, molten iron, etc. Casualty insurance premiums are going to be very pricey indeed.
If the tunnel were in a vacuum so that no heat was conducted into the train, it would still be nearly impossible to keep the train cool because of the high rate of heat transfer from infrared and visible light radiation as the interior temperature is expected to reach almost the same as the surface of the sun.
3) How advanced would you need to be to make this work? You would have to have force fields far in advance of star trek. We don't even have a working theory as to any physics that would allow for force fields. You need brand new physics. I.e., you need tech that seems like magic to us.
Assuming no friction, for the entire 38 minute duration of your trip, you are in free fall. You accelerate down until you reach the center at max. velocity of 28,440 kph and you start decelerating until reach the surface at zero velocity. At the exact center there is no acceleration and you are in a micro-gravity environment, i.e., your head would experience a small tidal force (a few milli-Newtons) pulling it down and your feet would experience a pull in the opposite direction but your net acceleration would be zero. Of course, no matter where you are in the vicinity of the Earth you experience similar tidal forces.
A final complication, antipodal locations are relatively rare that have land on each antipode. Only 4 percent of the earth surface terminates at land on each end. For example, Africa has no land mass at the antipodal locations except for some Pacific islands. Europe, basically nothing but a small fraction of France that has New Zealand as an antipode.
For a minimal comparison, consider the complexity of actually drilling the train tunnel under the Swiss Alps - recently completed and expected to open for service in 2016. Watch the video. This is nearly state of art drilling without the complications of the temperature and pressure you will encounter in a gravity train tunnel. Sorry about the annoying dramatics on the video, could not find a straight up quality documentary. I am also reminded that the Alaskan Way Viaduct has been very troubled (broken down digging machine). Digging tunnels is hard, even close to the surface.
Other than the fact that it is totally impossible for the foreseeable future, it would be a very interesting and efficient ride.
This article has nice explanation of the actual fall including the Coriolis effect that I neglected when describing the trip as free fall. @lorenpetch noted the omission.
The full truth is that I am not 100% certain of the real effect. Nearly all articles that refer to gravity trains simply refer to it as free fall, but I am pretty sure that there should be the Coriolis issue. So, I think Wikipedia, etc. is wrong, but did not want to invest the time to prove this to myself. Any takers? If you can do this rigorously it would be real nice of you to update Wikipedia too. It seemed beyond the necessary scope of this answer so I omitted it initially. The reason I am so nearly sure is because of angular momentum; I can't see any way to explain away this problem. So, I use the dodge of lateral acceleration coming from the train track (that does not affect the vehicle speed, just its path, keeping it in a straight line).
In a tunnel from pole to pole the Coriolis effect would not be an issue.
Continental drift and internal flows within the Earth would certainly complicate the tunnel, making expensive maintenance on ongoing problem.