What is the role, if any, of the Central Limit Theorem in Bayesian Inference? As someone who started by studying classical statistics where the Central Limit theorem is key to making inferences, and only later now am studying Bayesian statistics, I was late to realize that the Central Limit Theorem has a much smaller role to play in Bayesian statistics. Does the Central Limit Theorem play any role at all in Bayesian inference/statistics?
Later Addition:
From Bayesian Data Analysis by Gelman et.al. 3rd edition - on the central limit in the Bayesian context.
"This result is often used to justify approximating the posterior  distribution with a normal distribution" (page 35). I went through a graduate course in Bayesian Statistics without encountering an example in which the posterior was approximated with a normal distribution. Under what circumstances is it useful to approximate the posterior with a normal distribution?

The Frequentist needs asymptotics because the things they are interested in, like intervals which cover the true value 95% of the time or tests which have a false positive rate of less than 5% when the null hypothesis is true, typically do not exist. If the model is linear and the errors Gaussian, we can get exact confidence intervals, but rarely otherwise. However, we can build intervals which cover the truth asymptotically in very broad classes of models by exploiting a quadratic approximation of the likelihood.
The Bayesian does not have this problem. Given a prior and posterior, the 95% credible interval is a very well defined concept: any interval which contains 95% of the posterior mass. Likewise, Bayes factors can be defined in terms of posterior quantities. Life is easier in the linear/Gaussian case because these quantities will be available in closed form. But even in the general case, we can precisely define these quantities mathematically, and thus use the tools of numerical analysis to compute approximations. Most prominent would be Markov-chain Monte Carlo. The Bayesian, given infinite computing power, can thus get arbitrarily close to "correct" credible intervals/posterior means/etc for any sample size and any model.
[Of course, if the prior is not good, these quantities are utterly meaningless. Even if it is, they do not have any guarantee of relating to anything in the "real world" like a frequentist interval does; they are simply the results of "thinking rationally".]
You also ask about how Bayesians might avail themselves of the CLT. This comes in handy if the Bayesian doesn't have infinite computing power. MCMC is guaranteed to work eventually, but it might take too long on your computer. If the reason the posterior is expensive to evaluate is because you have a lot of data, we can deploy a normal approximation to the posterior. Various ways exist to choose the parameters of the approximating normal; perhaps the most popular is the Laplace Approximation, which uses a quadratic approximation of the posterior near its mode (this might remind you of frequentist asymptotics).
