It turns out that it's not too difficult to construct a counter example if you restrict the hyper-parameter space of the family of prior distributions. For example, let the likelihood, f(x|theta) only take on two values of theta, so the prior just puts mass p on theta=0 (i.e. P(theta=0) = p )and mass 1-p on theta=1. If you restrict p < 0.5, then the posterior will yield a distribution on theta with p > 0.5 for some likelihoods and some values of x.