As it's mentioned in the title. I am going to receive a formal letter this week. During my meeting with HR, they told me the visa will take 12 weeks so we can agree on the start-date at the point. My question is this: I really want to be successful in my postdoc and I want to do research as soon as possible, can I email tell the PI about this? (to get information about their current research step, etc.) or I should keep doing research and reading on my own and wait until everything is official?
[Disclaimer: I am not a lawyer, this is not legal advice]
As Thomas mentioned in his comment, generally, without a visa you cannot be legally employed in the country of the university, this includes remote work from abroad. In addition to that, at least in some countries, it is illegal to do work that is normally paid, even if you are not getting paid for it, unless you have a proper work permit. I guess, the logic is that you enter the labour market by doing work and not by getting paid.
So be careful and precise in your requests. Starting working on the project remotely, before your visa is ready and valid, is probably against the law (and your future boss will share some responsibility for this). However, simply asking for suggested reading before you started working on the project itself is probably okay. I suppose your PI would be happy to give those suggestions even if your visa fell through or you hadn't signed the contract to begin with.
A visa allows you to live in a country and (for the visa I assume you are getting) to be gainfully employed in that country -- that is, to draw a salary.
You do not need a visa to work on things while you are in your home country (working in your spare time, or in fact in a paid position). In other words, you can work on whatever project you'd like even before you get the visa. You just can't move to the target country yet, and you would have a separate conversion with your PI about getting paid if that's what you'd like.
This sounds like a case where legally and officially, you cannot be working as a postdoc before you receive the visa. Unofficially, no one is stopping you from working on a specific topic, or with unpaid collaboration, assuming you want to do this. If your aim is to continue research, such as to boost your publication record, most PIs would be willing to offer advice for getting started.
However, any work you do should be purely voluntary on your side (you can always say no to work until you are officially employed).
Please note one point: a visa is required to perform work (paid or not is not that relevant) in a certain location in the world.
Contrary to what the majority seems to think, if you are working remotely, you still need a visa, depending on where you are based and NOT on where your employer / client is located.
In other words, you can work remotely from the US for a Chinese company, as long as you have an US visa. Similarly, you can work from India for an US company, as long as you have an indian visa ... come on guys (&girls), the whole outsourcing business is based on this cornerstone!