Multicollinearity will certainty effect the inference you make. However, unlike in frequentist settings, the sampling process will not blow up. Instead, the ambiguity inherent in estimating coefficients of collinear predictors will show up as dependence in the posterior. See here for an example in which 1 predictor is entered into a model twice (e.g., yielding 2 maximally collinear predictors). Estimation goes just fine, but there’s lots of ambiguity about the values of these parameters (as there should be).