As we've already determined, Curiosity should last about 14 years on its nuclear power.
What happens near that timeframe? Will NASA start to manage the rover's use of power by shutting down unused tools?
It will survive the way that pretty much all space missions survive.
First of all, most space missions are speced such that the power at the end of the mission will meet 100% of the demand of the spacecraft. The mission of Curiosity is 2 years, thus, the spacecraft will have 100% of required power after 2 years.
Beyond that, its power will decrease. However, Curiosity has two batteries which will allow it to store charge, allowing it to still perform high power tasks, just for less time than it otherwise would be able to. So what will happen is the amount of moving, science, etc will be decreased over time.
The half life of the Plutonium 238 is 87.7 years. In effect, that means that the amount of power will only be halved every 87 years.
In practice, the limiting factor will probably be the lithium ion batteries, and not the RTG. But the effect will still be the same, you can't use your instruments as much, or move as far, but you can still do science to some extent, as time continues.