Visa preparation for the postdoc will take weeks. Can I start research (with the PI?)
[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.