Ohm's law is completely irrelevant to this situation *in the form you describe*. "Burning a hole through the board" would be possible and a simple function of Ohm's law only if they were using a linear regulator to generate the Vcore. But VRM's have been switching DC/DC converters since the 486 days. They achieve a voltage conversion by switching the incoming voltage on and off *very fast*, which results in an output voltage that's a function of the input voltage and the duty cycle of the on/off switching. An inductor (current-smoothing) and capacitor (voltage smoothing) give a nice clean DC voltage.
The differences between on-motherboard VRMs and this new in-package (it's technically a separate chip...) are significant. First off, physically moving it closer means that you're not sending 100+ Amps of current over the 3-4 centimeters of generally very thin copper traces on the PCB, they're sent millimeters through die-bond wires, or even through a solid substrate (no idea what Intel does at that level). There's your Ohm's law coming into play at that level, but the power losses there are relatively minimal since you're talking maybe a few tenths of an ohm. Die-bond wires are going to drop that to 10's of milli-ohms probably, so nothing major but still a positive effect.
The main reason this will generate a lot less heat is because of the *frequency* of the switching. Because this on-board VRM is so much smaller, it can switch the input faster (shorter wires, less parasitic capacitance, less ringing, etc.). This in turn means smaller value components required, e.g. the switch from the monster inductors seen on the motherboard (at maybe 1-2MHz switching) in the slide to the tiny chip-scale inductors on the FIVR (at 10's or 100's of MHz). The end result of all of this is that switching losses get significantly smaller. It's those losses that create heat local to the regulator. If they can for example go from an 80% efficient VRM to an 90% efficient FIVR for a 100W CPU load, they reduce the switching losses from 25W to 11.1W.