Restrictions on transitions in Spectroscopy Just for the sake of an example, let's assume a transition from one energy state to a higher energy state in atomic absorption spectrum

If the energy difference between the ground state and first excited state is E and that between ground state and second excited state is 3 E (just for say...). Now if an incident photon has an energy of 2 E so; 

(Assuming the electron is initially in ground state)


Will the electron show transition to the first excited state and emit a photon of energy E


OR


Will the electron not show any transition and not accept the photon at all, since the energy of the photon does not exactly match with the energy difference between any two specific states?


The example of atomic absorption spectrum is taken up just for the sake of explaining the question, the question is rather general and pertains to any kind of spectroscopy experiment in which their occurs transition from one state to another by absorption of EM waves.
If the energy is to be transferred from the photon to your atom one energy gap must match the energy of the photon. Thus absorption may occur from the first to second excited state in your example as the energy in both is 2E. However, there has to be some population in the first excited state for this to happen, zero population zero absorption. In fact it is a bit more subtle in that there has to be a difference in population between the two levels involved. It is possible for the photon to stimulate a transition from the upper level to the lower one, just as from the lower one to the upper one. The reason why this seems odd is that in most cases the population in the upper level is vastly smaller than in lower ones (via Boltzmann distribution) and the population in the upper level can often be set to zero without serious error. This is particularly true in electronic spectroscopy, as in the case of atoms, but far less so for pure rotational transitions in molecules which occur in the microwave region of the spectrum. You will find details in any good spectroscopy book.

Thus neither of your cases occurs but something different; transitions between the first to second excited state. How strong this transition is depends on the population in the two levels and on selection rules involving angular momentum and electron spin changes.