TLDR: There will almost always be correlation in the parameters in the Bayesian world; even if you use independent priors. Correlation might affect mixing rates, but that is going to be on a case by case basis. In general, you want to ask, "how can I account for the correlations"? Multivariate estimators of the asymptotic covariance of your estimators let you do that.


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There are two "correlations" between parameters that are present in this scenario (and in general in most MCMC scenarios)

 1. Correlation between parameters in the posterior. So the true covariance in the posterior is not a diagonal matrix. 
 2. Lag correlation across parameters as a consequence of MCMC sampling.
    So $Cov\left(\beta_1^{(1)}, \beta_2^{(1+k)}\right)$, etc. (Look at a cross-correlation plot to see how significant these lag correlations are).

Just like in frequentist settings, correlation does not necessarily impact the point estimates for the posterior means, but it affects the quality of the point estimate. So if $\mu = (\beta_0, \dots, \beta_p, \sigma)$ is the $p+2$ dimensional vector of interest, and you obtain $N$ MCMC samples ($\mu_i$), the point estimate is
$$\mu_n = \dfrac{1}{N}\sum_{i=1}^{N} \mu_i. $$

The two correlations mentioned above affect the quality of this estimator, since due to the Markov chain CLT,
$$\sqrt{n}(\mu_n - \mu) \overset{d}{\to} N_{p+2}(0, \Sigma). $$

Interestingly, $\Sigma$ breaks up nicely and explains exactly the two correlations mentioned above
$$\Sigma = \underbrace{Var(\mu_1)}_{\text{Posterior Covariance Structure}} + \underbrace{2 \sum_{k=1}^{\infty} Cov(\mu_1,\mu_{1+k})}_{\text{Covariance due to correlated samples}}. $$

Just like in the usual MLE setup, if you can account for $\Sigma$, you account for all the correlation in estimation process. Recently consistent estimators for $\Sigma$ have been proposed, and you can now actually use these to say, "well this is the amount of error in the estimator, so do I have enough sample?". The R package [mcmcse](https://cran.r-project.org/web/packages/mcmcse/index.html) lets you estimate $\Sigma$. You can also use functions called multiESS and minESS in it to find out how many effective samples you need, and what your effective sample size is. These calculations are done using estimates of $\Sigma$, and thus account for the correlation.

[This](http://arxiv.org/abs/1512.07713) paper explains in detail.