Can an independent institution run experiments without IRB approval? Lets say a rich person has the funds to create a US-based research institution that does not receive any form of public or private funding with the exception of said persons bank account. Will this institution be able to run experiments on human subjects without IRB approval? Nothing extreme like giving people new medicine or trying out novel surgery techniques but rather something mundane like collecting surveys or conducting psychology experiments.
At which point does it become illegal to run an experiment with no oversight within US jurisdiction?
I'll address the academic, rather than legal aspects of the question.  All legitimate scientific journals require ethics review for published human subjects research.  Any human subjects research which is has not passed ethics review must not be published.  In my opinion, any such research is not part of main-stream science.  So the answer is no.
It is generally accepted that these are not research, and therefore not human subjects research:

Testing a commercial product to see if customers like it.
Experiments whose results will be kept secret.

However, all activities need to be ethical, even if they are not subject to IRB requirements.