I did an unpaid postdoc in Japan, as my wife was staying there anyway, and I was both young and desperate. I knew of 3 or 4 other people like me, also motivated by a desire to stay in Tokyo at "all cost". I wouldn't say they were common, but they weren't unheard of either.
The rationale was that I would get to publish and given the prestige of the university and the lab I was with, it would be beneficial for my career in the long run, even if I wasn't paid. I signed on as an unpaid postdoc and I made ends meet by teaching English, French and translating technical papers and documents.
My experience was overall negative. Despite being reassured otherwise - I was told I would be treated just like any junior faculty/researcher - people just don't take you seriously if they know you are not paid, and it will be very hard to keep it a secret. The psychological effect became very difficult after a while. I expected most people to admire my perseverance and dedication to my research, by willing to work for free. Instead most people were looking at me as some sort of charity case - "he's working here for free because he can't get a real job". There were several conflicts between my research (what I wanted to be doing), and the part time jobs I was doing. It was very hard to focus on my research, when I was constantly being pulled into some task or event because of my other jobs.
If you are a foreign national in the host country you want to do a postdoc in, you will most likely run into visa issues if you are not paid, and you will run into employment authorization issues for whatever part-time work you want to do to make ends meet. In my case I had to switch from a visiting scholar visa to a dependent visa, then wait a long time to get a part-time work authorization. I've seen foreign researchers in the US (where I live now) face similar problems.
In the US, some states don't allow unpaid post-docs for labor law reasons, and even when it is allowed by local laws, the university itself might have rules against it for security, insurance and liability reasons. The 3 places I am familiar with in the US, Georgia Tech, Emory, and MIT, don't allow unpaid post docs.
Back to my experience in Japan, none of the unpaid post-docs I knew of materialized into real academic positions. They all eventually gave up, and either went back to their hometowns/countries and got low-level teaching positions there or they left academia for the industry - and when you join the industry you find yourself having lost 3 or 4 years compared to other people with your age and skill set.
Conclusion: Don't go for an unpaid post-doc, it's very taxing and it's not worth it at all. There's a world beyond academia.