MCMC can be used for Bayesian inference of other models with hidden variables. Gibbs sampling, for example, is used in Hidden Markov Models. Here is a paper that discuss the differences between MCMC and the more classical approach using the EM algorithm.
The question is: Are HMMs a useful model in finance? Some academics argue that they have predictive power.
One can look at: Stock Market Forecasting Using Hidden Markov Model: A New Approach. I'm not convinced by the approach they use.
On the other hand, HMM can be used to build volatility filters for trend following strategies.
There are certainly other models parameters that can be inferred using MCMC. I personally find it very time consuming (this is only based on experience and not on convergence analysis). Furthermore, as stated in the first paper, if one wants to use Bayesian inference then the EM algorithm can be used for computing MAP parameters.
All in all I haven't found it very useful.