Why are there recommendations against using Jeffreys or entropy based priors for MCMC samplers? On their wiki page, the developers of Stan state:
Some principles we don't like: invariance, Jeffreys, entropy
Instead, I see a lot of normal distribution recommendation. So far I used Bayesian methods that didn't rely on sampling, and was kind of happy to have understood why $\theta \sim \text{Beta}\left(\alpha=\frac{1}{2},\beta=\frac{1}{2}\right)$ was a good choice for binomial likelihoods. 
 A: They do not provide any scientific/mathematical justification for doing so. Most of the developers do not work on this kind of priors, and they prefer to use more pragmatic/heuristic priors, such as normal priors with large variances (which may be informative in some cases). However, it is a bit strange that they are happy to use PC priors, which are based on Entropy (KL divergence), after they started working on this topic. 
A similar phenomenon happened with WinBUGS, when the developers recommended the $Gamma(0.001,0.001)$ as a non-informative prior for precision parameters since it resembles the shape of the Jeffreys prior. This prior became the default prior for precision parameters. Later, it was shown (by Gelman!) that they can be highly informative.
A: This is of course a diverse set of people with a range of opinions getting together and writing a wiki. I summarize I know/understand with some commentary:


*

*Choosing your prior based on computational convenience is an insufficient rationale. E.g. using a Beta(1/2, 1/2) solely because it allows conjugate updating is not a good idea. Of course, once you conclude that it has good properties for the type of problem you work on, that's fine and you might just as well make a choice that makes implementation easy. There's plenty of examples, where convenient default choices turn out to be problematic (see Gamna(0.001, 0.001) prior that enables Gibbs sampling).

*With Stan - unlike with WinBUGS or JAGS - there is no particular advantage to (conditionally-)conjugate priors. So you might just a well ignore the computational aspect somewhat. Not entirely though, because with very heavy tailed priors (or improper priors) and data that does not identify the parameters well, you run into problems (not really a Stan specific problem, but Stan is quite good at identifying these issues and warning the user instead of happily sampling away).

*Jeffreys's and other "low information" priors can sometimes be improper or be a bit hard too understand in high dimensions (never mind to derive them) and with sparse data. It may just be that these caused trouble too often for the authors to never comfortable with them. Once you work in something you learn more and get comfortable, hence the occasional opinion reversal. 

*In the sparse data setting the prior really matters and if you can specify that totally implausible values for a parameter are implausible, this helps a lot. This motivates the idea of weakly-informative priors - not truly fully informative priors, but ones with most support for plausible values.

*In fact, you could wonder why one bothers with uninformative priors, if we have lots of data that identifies the parameters really well (one could just use maximum likelihood). Of course, there's plenty of reasons (avoiding pathologies, getting the "real shape" of posteriors etc.), but in "plenty of data" situations there seems to be no real argument against weakly informative priors instead.

*Perhaps slightly oddly a N(0, 1) is a surprisingly decent prior for coefficient in logistic, Poisson or Cox regression for many applications. E.g. that's very approximately the distribution of observed treatment effects across a lot of clinical trials.

