What are the requirements for a non-citizen to open a bank account in the United States?

For regular checking/savings account, a Tax Number is not required for opening an account.

Most US chartered banks require the following:

• A valid ID (for foreigners, this is clearly the passport). They will make copies of it, as well as the Visa &/or Arrival Stamp page. Some and I stress some, may scan the whole passport.
• They will ask you for a secondary identification (it can be a credit card, driver's license, debit card, ATM card, any other form of identification).
• Some banks can request for a picture based second form of identification only, but passport is usually good enough.
• A local address is required and you will have to provide some form of a bill, etc. attesting that to the fact that that is your local address. If you are forwarding email to a friend, etc. They can ask you for your friend's proof of residence to that address.
• Proof of funds (if depositing cash in large amounts).
• Proof of funds if you are wiring funds in.
• They will check their own internal KYC list to see if anything else is required
• They will confer this with their manager or supervisor (most US banks will do the 4-eyes, i.e. two people signing off on the account opening OK).
• If you are legally staying in the US for extended period of time (like student, etc.) a copy of your I-20 etc. would be required
• If you happen to be from a few countries of interest (essentially any Arabic, Muslim country, etc.) you will be subjected to a more thorough examination of your documentation (this is the silent rule, as much as you may deny that it doesn't happen).

That's about it.