How to get the phase response of an arbitrary LTI system?
I'm assuming you are trying to get a bode plot which shows the magnitude and phase response you would get if you were to test the the system by slowly sweeping the frequency a constant amplitude sine wave input and measuring the output amplitude and phase at various points.

If this is the case then there is only one frequency \$ \omega \$.  Now we know that \$ e^{j \theta} = \cos(\theta) + i \cdot sin(\theta)\$.

You can therefore treat each exponential term as a vector with amplitude and phase and add them as vectors.

Note however, that if you compare the results with a real digital filter you will not get exactly the same result because the digital filter is only taking measurements at certain instants in time (the sampling frequency) and the data is quantised.  However providing the sampling frequency is much larger than the frequency of interest it is a reasonable approximation.