Requesting feedback from a postdoctoral application
Asking for feedback on your application from someone who has turned you down is generally unlikely to yield useful information, and it may annoy them by making them choose between saying something unsatisfying or not replying.  Here are some of the issues that have run through my mind when I've been asked for feedback (which is not common but happens every once in a while):


I wonder whether I'm dealing with someone argumentative or difficult, who will try to dispute my explanation or change my mind, use the feedback as evidence in an accusation of misbehavior or dishonesty, or misquote me to others.  Even a perfectly reasonable applicant might feel upset if they re-apply next year and get rejected again after having felt they addressed all the issues I raised.
In most cases there's nothing objectively wrong with an application.  All I can say is "your application was good but we liked someone else's better", which is not informative.
Comparisons with other applications are crucial, particularly with the successful application, but all the details of other applications are confidential.  It's difficult to explain why we liked someone else's application better without revealing anything that isn't already public knowledge.  There won't be a simple quantitative explanation ("you should have published two more papers in high-prestige journals"), but rather a subtle qualitative comparison that is difficult to summarize in the abstract.
Furthermore, letters of recommendation are one of the most important parts of an application, and their contents should not be revealed to the applicant.  It's tough to give useful feedback without touching on the letter contents, especially because most concerns I might mention would at least make it clear that the letters didn't resolve these concerns.
In some cases, I can identify a problem with an application that is not based on letters of recommendation or direct comparisons with other applicants.  However, these problems are often not easily fixable (e.g., "we just didn't find your paper topics interesting").  Pointing them out feels like adding insult to injury.  It might be useful for the applicant to hear that they never had a chance at this job, but I don't want to be the one to say so.  The best case scenario is that the applicant always remembers me as the person who delivered this insulting news, and the worst case scenario is that I make the committee look foolish if we misjudged the applicant.


It's only in rare cases that I have genuinely useful feedback, something fixable that I can ethically reveal to the applicant and that might make a real difference in the future.  (This usually indicates that either the applicant's advisor is negligent or the applicant is not listening to advice.)  I've never had someone ask for feedback in this situation, although it might happen someday.  In a few cases I've tried to communicate this information through a backchannel, such as one of the applicant's letter writers.

In your case, there's almost certainly nothing seriously wrong with your application on paper, or you wouldn't have been interviewed.  Assuming you didn't do anything dreadful during the interview, you presumably fall into the second case above (your application was good but another was even better), and I doubt anyone will be able to give you satisfying or useful feedback.

If you decide to ask for feedback, I'd recommend indicating that you are aware there might be little or nothing to say.  For example, you could send a thank you e-mail and include something along the lines of "I realize that you may have no constructive feedback to offer beyond that another candidate was an even better fit for the position.  However, should you happen to have any advice for how I can strengthen future applications, I would greatly appreciate it."  This relieves the pressure by giving them an easy way to decline to say anything substantive, while bringing up the topic just in case.  I don't think there's any need to do this, or that it is likely to result in useful feedback, but it can't hurt.