Being burned at stake would count as "having one's liberty taken away", and is therefore a case where, as I said, "innocent until proven guilty" is the appropriate standard.
Why is it the appropriate standard? Because as a society, we believe that taking away an innocent person's liberty is far worse than letting a guilty person go free. For the expected value (in the statistical sense) of the legal system's benefit to society to be positive, the prevalence of the former should be a tiny fraction of the prevalence of the latter. (How tiny that fraction actually is, in the US or anywhere else, is debatable - you could look at the number of convicts who were proved innocent upon the advent of DNA testing - but that's not the point.) Thus we should only convict if the apparent probability of guilt is extremely high.
But if I as a bystander believe an innocent person is guilty, it's not as big a deal. True, if a lot of people believe that, it causes some reputational damage, which can be bad. If someone with a connection to the person makes decisions disfavorable to them based on that belief, that's quite bad. But this risk is mitigated by the fact that soon enough the police will be required to present their evidence in court, which should help drive our assessments with more certainty to one possibility or the other. And in any case, these harms are far lesser than the harm of subjecting the person to prison, or worse punishments.
Perhaps I misstated the ideal standard for bystanders. Perhaps it should not be literally whichever possibility is more likely; perhaps we should give people some "benefit of the doubt", due to the above harms. But there's no need for the extreme standard demanded by the actual justice system. A reasonably high probability of guilt is enough, at least to treat guilt as a working hypothesis.
In a legal system such as the United States', "if the police took him" (specifically, indicted him), I'd say there's a high probability - at least 75% or so, probably higher - that there's a "good reason for that to happen". (At least in the sense of "he did what they say he did"; whether that thing ought to be illegal or not is typically more subjective.) It's not certain; there could be some sort of corruption or unethical behavior by the police, or they could simply be mistaken. But it's enough to form a working hypothesis, to use until we gain more information.