To do a private certificate, IR or multi, you would need to go through the TSA process which require visa details.
It's been discussed here on pprune before and the general consensus of the population here was that it's not the TSAs job to verify you have a visa. In fact, if you look at the original ruling on the TSA process (a very dense PDF file on the TSA website) you'll find that the word "visa" is only mentioned twice or so, in a completely different context.
It is true that the TSA, as part of the information gathering on you, ask for your passport number and visa details. But as far as I know, the site allows you to specify "no visa". (Mind you, it was three years ago that I went through this process and this is something that I don't remember clearly, plus the site has been changed in the meantime.)
But if the TSA *requires* you to have a visa somehow you can point them to their own regulations and say they're wrong. Particularly if you can prove that you do not need a visa according to the INS for whatever you are going to do.
All we need is a brave soul to test out this theory though.
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To add, I just browsed the AFSP website, and here's what the website itself has to say about this on the pages where you submit your personal information:
U.S. Visa: If you have a U.S. Visa, please enter this information. Do not list visas from non-U.S. sources.
* AFSP may cancel an applicant's flight training request if AFSP becomes aware the candidate is intending to take flight training without the appropriate immigration status. If you have questions about a cancelled request, you may call AFSP at XXX
So it looks like the TSA simply follows INS rules on whether a visa is required or not. There doesn't seem to be a blanket visa requirement for all cases.