None that I've dealt with have required that. They
do require to verify that you are who you claim you are, and without a US address you might find that requires extra paperwork (such as getting official proof of address and identity document certified by a US consulate or a notary public), and some banks might just not want to deal with the hassle. You also do need to fill out forms to establish your tax withholding.
I've held personal and business accounts with Chase and TD Bank, as well as trader accounts with a couple of brokerages, and the documentation requirements were pretty basic in all cases. I've looked into opening accounts at other banks at various points too.
In some cases the easiest alternative is to simply buy an off the shelf US corporation, and have the company that sets it up register a US bank account for it. That's certainly often less paperwork and hassle than dealing with a personal account, especially if you set up a Delaware corporation.