I've been following closely on the tragic situation with the British tourist, Grace Millane. This case hits pretty close to home for a number of reasons (one of which being because it's physically close to where I live).

One claim I saw in an article I was just reading on NZ herald by a retired police officer, was that mobile phones can be tracked even when turned off (after the fact). How much merit is there to this? 

I know there's been discussions of the police being able to turn on location tracking remotely when 111 is dialled. Cellular triangulation is also possible outside of that system (technically speaking, legalities aside). 

Now I have seen claims that certain mobile devices can be turned on remotely, though I've never see clear evidence of that being actually true. But I have never heard of tracking a phone, after the fact and certainly not while turned off.

Is this something that is technically possible? 

My understanding is that phones are Off when the are turned off. No radio's are active, including GPS. It would require hte phone to be turned back on and registered against/near a cell site. 
The only time I could see GPS being possibly useful is if the phone was turned on, out of cell coverage and not in flight mode, in which case a lot of phones provide location tracking. But then this would require physical possession of the phone, or for the information to later be uploaded somewhere.