Shine is defined as to give out a bright light or to glow or be bright. Diamonds do, in fact, reflect light, thus according to the definition, they do shine.
Diamonds are expertly cut and polished to maximize a their natural brilliance. Reflection, refraction and dispersion are the three main factors involved in diamond brilliance.
The cut of a diamond helps determine the brilliance of the stone. Cuts enhance the brilliance of a diamond, and shapes are designed to take advantage of the best light performance. If symmetry is off by a fraction, light will not reflect properly.
The instantaneous shine when a light hits a diamond is called reflection. Only a portion of light hitting a diamond reflects back, while the rest passes through it. Refraction is light moving through a diamond, fracturing and scattering, causing the well known diamond sparkle. The bouncing light inside a diamond causes a prism, or rainbow, effect called dispersion, and adds to the overall shine of a diamond.