The iPhone 2.0 software - available for the iPhone 3G, iPhone and iPod touch - can't automatically turn any email account into push email. It can only instantly send new messages to your phone if your email service supports push email.
The new iPhone 2.0 software supports push email in conjunction with Microsoft Exchange servers and Apple's MobileMe service (the replacement for Apple's .Mac). Apple has labelled MobileMe "Exchange for the rest of us" and it can push email to devices running iPhone 2.0, plus keep the device in sync with Microsoft Outlook on a PC or Mail, Address Book and iCal on a Mac.
The new iPhone's main settings menu now features a new icon - "Fetch New Data" - which lets you enable push email. You can also schedule the phone to automatically check non-push email services every 15, 30 or 60 minutes, or else only manually.
Which email works with push?
Push support from other services depends on the provider. The original iPhone supported push email with Yahoo!, by routing email through imap.apple.mail.yahoo.com, but the service was flaky. This is still the default Yahoo! email wizard configuration under iPhone 2.0, but if it refuses to work you can still go into Other and manually set up POP3 Yahoo! access. Gmail does not support push email, only IMAP or POP3.
What if I'm using Gmail, or my email from an ISP like iiNet?
Of course the workaround is to forward your existing email address, such as a Gmail account, to your .Mac account. You could even create server-side filters on Gmail to only forward emails from certain senders to your .Mac address, to ensure you're not bombarded with push email spam.
You can also configure MobileMe to automatically check external POP3 email accounts, but this won't get emails from a Gmail account to your phone as quickly as it would if your Gmail account was automatically forwarding them.
Push email uses more data
The iPhone 2.0 mail client only downloads email headers from IMAP and POP3 servers, in order to save bandwidth. When accessing push email services, the phone automatically downloads the body of the email and small attachments - meaning you need a generous monthly data allowance.
The iPhone disables its wifi adaptor when sleeping, switching over to the mobile data network. This means most of your push email will come down over the mobile data network while the phone is at rest, even when you're within range of a wifi network.