Let's say we have a blue, opaque material. If white light was incident on that material, the blue light would be absorbed by electrons and the electrons would transition to a higher energy state, and then the electrons would transition back to the original state (ignoring any other states it could drop to) re-emitting that light to make the material appear blue, but what happens to the rest of the light? Is the rest of the light absorbed by molecular bonds and converted into heat?
I'm not sure what you mean by "the rest of the blue light." Basically three things can happen when low energy light passes though matter.
Absorption and reemission. The light excites a dispersion free mode, like your electronic energy state example, in that case the light will be reemitted at the same wavelength most often. You can work out the exact distribution with the Bohr-Dirac model (bohr model with relativistic corrections).
Absorption and dispersion. Light can excite phonon modes, those will disperse and heat the object up.
Finally, light can pass through the material and not interact with the object at all.