Do unethical acts that are tangential to methodology taint research?
First, I think there is an important distinction to be made between unethical and illegal. 

While most things that are illegal are also unethical (e.g., murder, theft), there are many cases where the laws themselves are either clearly unethical (e.g., slavery) or may be considered unethical by a reasonable person (e.g., prohibitions on stem cell research).

Let us thus consider only unethical acts unrelated to research, whether illegal or not.  Here, I think there are at least two good motivations for considering unethical acts that do not directly call the methodology into question:


Concern about honesty: If you find out that somebody has been gratuitously lying in other aspects of their life, it's entirely reasonable to wonder whether they would feel comfortable lying in their research as well. Motivation matters here: a person lying to protect themselves or to cover a sexual infidelity is not particularly relevant, while a person lying with the intention of advancing themselves or hurting others is of much more concern. In this case, there is no "smoking gun" indicating results should be withdrawn, but it would be entirely reasonable to do a careful process re-review of their work for validity in the light of the new knowledge.
Punishment: If a person does terrible things that damage the scientific community or the larger community, one may wish to exile them and their research as part of the punishment, in order to make it clear one should not expect to profit from such behavior. This is clearest in the case of unethical acts directly related to research, such as sabotaging a rival or non-consensual medical experiments, even when they don't invalidate the methodology and conclusions. It's less clear what one should do about the scientific legacy of say, an abusive rapist who happens to also be a highly successful mathematician.  

There is still no clear consensus on what to do in such cases, even in the case of Nazi medical research.  Personally, I favor the approach suggested by some of using only when no other citation is possible, and then without a formal citation and with inclusion of an explanation of why this research is problematic. Should this be extended to lesser criminals like rapists and abusers?  Unclear, though I think it's certainly worth considering for those who poison a scientific community with their behavior.