How could have Banksy kept an RF receiver running for 12 years?
A relatively large lithium primary cell would be my choice. They are specified for something like 10 or 20 year life running water meters, including periodic radio communication. And maybe a second cell to run the motor so it stays relatively fresh. 

The Israeli company Tadiran makes such products. 

As long as the temperature does not get too high the shelf-ish life of such Lithium cells is in the decades (they claim 40 year operating life). You can bet that an expensive piece of art will be kept in carefully controlled temperature and humidity conditions.  

Given that power source, power management would be important but not crazy critical. You would want to keep the average draw in the &lt;&lt; 100uA range most likely. Doing the sums, their 19AH TL-4930 works out to 54uA (average) for 40 years. So even if it turned on a receiver once per 10 minutes for a second it could draw tens of mA, provided the sleep mode power was inconsequential.