I don’t know about the visa, but even aside from that there are legal, ethical and reputation issues with your idea. Considering all of those, I’m 99% certain that neither MIT nor any other reputable university will give even passing consideration to the idea of an unpaid postdoc.
On the legal side, there is this thing called minimum wage that forces organizations to pay their employees. The exceptions are volunteers and unpaid internships, and there are rules governing who can count as either of those, to avoid organizations using those categories as loopholes to exploit workers. I don’t know enough about this topic to be able to say for certain that a postdoc couldn’t be categorized as an unpaid intern, but I suspect that at the very least, the idea would be unusual and eyebrow-raising enough to deter anyone from wanting to even test whether this is possible.
On the ethical side, postdocs are already a somewhat vulnerable population that are paid less than their qualifications make them “worth” on the free market (outside of academia). This raises real issues about the potential for exploitation. There are reasonable arguments why this pay structure makes sense, but employing unpaid postdocs absolutely does not make sense, and would undermine the ability of people who can’t afford to take such unpaid positions from being competitive in the academic jobs market. The last thing a university would want is to open itself to accusations of such exploitative behavior.
On the reputation side, a university like MIT can afford to hire the best postdocs in the world, and its reputation is much more valuable to it than the cost of employing a postdoc. If you are good enough to want them to work for you, they will offer to pay you like any postdoc, to avoid anyone getting the impression that they are settling for subpar talent in order to save money, if for no other reason - even setting aside the legal and ethical issues I mentioned above.
Bottom line: an unpaid postdoc at a good university almost certainly isn’t going to be an option for you, sorry.