Continuous Time Signal and Discrete Time Signal - Connection Between Periodicity and Discretness
This is a 2 parts answer:


Given a discrete representation of the Spectral Data does necessarily represent a periodic function in the time domain?
Well, it is by definition. Look, the DFT is a fancy name to a scaled Discrete Fourier Series, namely combination of harmonic signals. By the nature they are all an integer multiplication of a base frequency. Now think about a linear combination of finite harmonic signals which all are multiplication of known frequency -> The are all periodic in the lowest frequency. And so is the signal.
Can all periodic signals be represented by a Fourier Series?
Now you are asking about convergence and there are few requirements about signals to be represented by a Fourier Series which deals with the ability to integrate them over the period. Another issue, when you say Fourier series, remember an infinite number of signals are considered.