Assuming the Moment Generating Function exists, $E[e^{tX}]$, and we evaluate it out is it always going to valid to just let $t=iw$ and get the Characteristic Equation?
For example with Normal:
$E[e^{tX}]=e^{\mu t+\frac{1}{2}\sigma^2t^2}$
$E[e^{iwX}]=e^{i \mu w - \frac{1}{2}\sigma^2w^2}$
Any examples of when this wouldn't hold true?
Yes. If $f(t)=E\exp(tX)\lt\infty$ for all real $t$ in a neighborhood of $0$, then $f$ is analytic in a neighborhood of the imaginary axis, and its restriction to the imaginary axis is the characteristic function: $E\exp(itX)=f(it)$ for all real $t$.