Do black holes ever go away?

Indeed, black holes will eventually evaporate. This is known because they release Hawking Radiation, which is produced by quantum mechanical processes close to the Schwarzschild Radius (event horizon). The radiation from the black hole will cause the hole to evaporate over an extremely long period of time until it vanishes completely.

In fact, black holes can lose mass and eventually evaporate completely through a process known as Hawking radiation, which was first described by physicist Stephen Hawking. This process happens when pairs of particles and antiparticles are formed near the black hole's event horizon. If one of the particles falls into the black hole while the other escapes, it appears as though radiation is emitted by the black hole. Nevertheless, this process is incredibly slow, and black holes are practically always thought to be permanent features of the universe.