What would happen if a supermassive blackhole at the center of a galaxy evaporates? Well the title says it all but I will just explain it in detail a bit. We know that a blackhole eventually has to evaporate due to the hawking radiation. It is also said that the blackholes at the center of galaxies are the galaxies' source of energy. So what would happen if that blackhole evaporates?
 A: The supermassive black holes at the centers of galaxies have evaporation times much, much larger than the age of the universe. They're not going anywhere.
But, even if they hypothetically vanished, mature galaxies would barely be affected. It's not quite accurate to say that these holes are the "source of energy" for the galaxy. The stars in the galaxy usually orbit and shine quite happily without feeling any impact from the black hole. 
However, when the black holes are undergoing rapid accretion, the radiation emitted from their disks is hypothesized (for sufficiently large black holes, not found in every galaxy) to drive away gas from the galaxy, which might prevent the formation of new stars. 
So, if we performed the thought experiment of traveling back in time and removing the black holes in very young galaxies that would have grown very massive black holes, then this line of thinking suggests that those galaxies in our experiment would form even more stars by the time they mature than we currently see in them. It's either that, or the galaxies wouldn't evolve very differently at all from the way we see them now.
A: This is really an extension to kleingordon's answer.
The net rate of evaporation of a black hole is the rate of energy loss through Hawking radiation minus the rate energy/mass is accreted from the universe outside the black hole. At the moment the temperature of a supermassive black hole is far lower than the cosmic microwave background, so even a completely isolated supermassive black hole will grow over time not shrink.
As time goes on the expansion of the universe will cool the CMB until it becomes lower than the temperature of the black hole, but this will take a long, long time. The temperature of the supermassive black hole at the centre of the Milky Way (Sagittarius A$^*$) is around $10^{-14}$ Kelvin so it will take around $10^{14}$ times the current lifetime of the universe before Sagittarius A$^*$ even starts to evaporate.
Now galaxies are not static objects. They evolve in lots of different ways, and after $10^{24}$ years they are likely to look completely different. So when you ask what effect the black hole evaporating will have on the galaxy, you need to bear in mind that the galaxy we know and love will no longer exist by that time.
