For example, if I define my prior distribution as Gaussian but I define my likelihood as multinomial, does this break any theoretical basis for variational inference?
No this breaks no theoretical basis for Bayesian inference, nor for Variational inference. However, it may make your computation more complicated. Depending on the form of variational inference this may be more or less of a concern. For example, in mean-field variational inference you will likely break the nice, closed form solutions you get when you choose the conjugate priors.