Is it common for professors to ask a postdoc to start working without compensation while waiting for paperwork to be completed?
Asking someone to work without compensation is illegal in many jurisdictions. It is also hardly ethical. Unfortunately, it is also not uncommon in academia.
In modern "publish or perish" academia professors are often under a huge deal of pressure from university administration to produce countless high-quality papers and teach ever-growing number of courses. A significant proportion of academics feel that they are forced to work extra hours to meet the expectations of their universities. This draining and insecure work environment often impacts the judgement of academics when they manage the work of their PhD students and postdocs. Many sincerely believe that they should "prepare" their students/postdocs to the hard realities of academia and make them more competitive by demanding to produce results at a cost of personal life and well-being. This is wrong, but unfortunately, wrong things do happen in this world and in academia as well.
UPD: Answering some questions in comments, I feel that I have to explain why the PI's behaviour is wrong. I don't think it is a subjective judgement.
- Forcing someone to work without compensation and without contract is modern slavery. Slavery is wrong. Postdoc are employees, they have terminal education degrees, they are professionals and they do important research work, not some glorified "training". Forcing a postdoc to work without contract and compensation is wrong.
- But perhaps, the PI was not forcing the postdoc, but merely asked them nicely whether they would like to volunteer? Well, let's see. Postdoc does not know whether they can say "no" to their PI without risking their contract or compromising the work relations with their PI during their contract. There are so many ways in which postdoc's whole career depends on their PI's opinion (access to research resources, appraisals, extensions of contract, letters of recommendation, etc). In this situation postdocs are under a huge pressure to say "yes" to unfair and illegal proposals. That's why even proposing to work without contract is abusive, morally wrong and often illegal.
There is a difference between no compensation and delayed compensation. An important question is how routine this paperwork is. If it is merely a bureaucratic hold up and you are already in town, why not get a jump start on your research? It could make the research when you officially start easier, and could lead to a better overall research experience, which could help your long-term career.
Hopefully the professor is enthusiastic about the research. From their point of view, this could be allowing you to start early rather than forcing you to do so. The only thing that would be a red-flag for me would be if you weren't given any choice. You should have the freedom to say something like "I would prefer to get settled while I'm waiting for the paperback. Can you suggest some background material that I might read in the meantime?" (or words to that effect, the business about reading at the end since you would still want to communicate a bit of enthusiasm for the project).
The underlying issue you're encountering here is that, generally speaking, a postdoc position is considered an education (albeit a paid one) for the position holder. As a consequence, telling a postdoc before they formally start their job that they might want to start reading some background material or working on the project (presumably while they have no other job because (i) they've graduated with their PhD and (ii) are waiting for their visa to process) is not so different from telling a student at the beginning of the summer break "Look, this is the material I'm going to cover in MATH 517, and here is the book we're going to use; why don't you start reading up on that material now already instead of waiting for the beginning of the fall semester?".
Strictly speaking, it might of course be illegal to have someone work for you while they are not paid. But graduate students are also not paid and still work on research projects -- the point simply being that it actually benefits both the graduate student and the future postdoc to already work on the project because they are also working for themselves (through publications or, more generally, for future job prospects).