Why DTFT coefficients are periodic and why continuous Fourier transform coefficients are not periodic?
The explanation by @Maximilian Matthé is a standard and formal approach for this question. But I think it is not intuitive and easy to understand the reason. In the following, I will try to explain from another aspect. 

First of all, periodicity means infinity. For a signal in the temporal/spatial domain, the periodicity is somewhat easy for realization, such as extending it periodically and infinitely. But for the frequency domain, it is quite different. Different position means different component with different changing speed. Therefore, if periodicity is expected in the frequency domain, infinitely rapid changing is expected in the temporal/spatial domain. In the real world of our life, most signals changes smoothly, thus periodicity is impossible in the frequency domain. While for discrete-time signals, the model is based on the $\delta(t)$ function, for which the changing is infinitely sharp and rapid, thus periodicity becomes possible in this case.