I don't know if this will have contributed to the burnout, but I have discovered that I'd got the pins all muddled.
The power and ground pins were fine, but I had misinterpreted the position of the nixie pins so that I had the number sequence reversed.
I had also failed to spot the utterly confusing layout of the logic pins on the IC. If you look you'll see that the pins are labelled A, D, B, C (in that order). The logic chart is labelled D, C, B, A.
I will have undoubtedly sent things to the IC that it wasn't expecting, though the logic chart says that should merely have resulted in no output.