I purchased a Kyosho Inferno VE RTR about a week ago.
First run I was just playing around in the driveway and all went well.

Second run I went down to a local road track with a mate (who also purchased a VE from you a week before), fitted road wheels, and hammered around for a long time (great battery life!)
Eventually the batteries must have been exhausted (2x 5200mAh LiPo) and the car braked to a stop. At that point I thought everything was fine.
Later I borrowed a pair of batteries from my mate and tried them out - motor wouldn't run for more than a clunk. I figured the batteries were flat - turns out I was wrong.
Anyway I went to try the car this morning, after charging my batteries last night.
Same problem - the ESC goes through it's startup sequence ok, but when I try the throttle, it dies. The ESC reboots at this point so there's obviously something wrong. This morning I had it connected for maybe 15s and it started making a strange noise - whining - so I disconnected everything. The motor, ESC, battery, and wires had all become hot in less than 15s!
I was wondering
(a) if anyone knows what might be wrong?
(b) why it might have happened - only second run of the car :-(

It uses a Team Orion Vortex 8 ESC/550 motor combo.
At the end of the run the motor seemed hot, but not incredibly hot.
It was a reasonably long run, mostly not at full speed (tight road track!) but nothing seemed very hot.

It sounds like your motor has an internal short. That will make the motor hot very fast along with everything else as it's pulling a lot of current. I would call Kyosho and see if they can send you another setup or if you can send it in as it should still be under warranty.
Good luck.