Suppose a photon collides with an electron, but does not have enough energy to dislodge the electron or knock the electron into it's next energy level. What happens? Is the effect observed?
Nothing happens. The probability that an electron exists between energy levels is zero, so the probability that it will absorb that level of energy is zero.
The photon will scatter off from its initial direction, transferring a little momentum and kinetic energy to the atom as a whole and losing a little bit of energy, thus becoming redshifted. It is like the Compton scattering, only that instead of the mass of a free electron, you should use the mass of the atom.