This can now be accomplished very easily in iOS 7.
(n.b. The following assumes that you have already setup your account on the iOS device. These steps described are for a generic IMAP account. Also, when ALL CAPS are used below, it is to reflect how Apple has it listed in iOS 7.)
- Launch the 'Settings' app on your iOS device
- Scroll down and then tap on "Mail, Contacts, Calendars"
- Tap on the email account you want to configure.
- Tap on the 'Account' line with the
> at the top of the next screen
- On the 'IMAP ACCOUNT INFORMATION' screen, scroll all the way down until you see 'Advanced' and tap on it
- Now you will see a 'MAILBOX BEHAVIORS' screen, showing you which mailboxes are used for Drafts, Sent, Deleted, and Archive mailboxes.
- Important: Tap on the 'Archive Mailbox' line and verify that you have chosen the correct folder on the server where you want messages to be saved.
- Once you have verified that the Archive Mailbox points to where you want it to go, look for the 'MOVE DISCARDED MESSAGES INTO' section, and tap 'Archive Mailbox' (instead of 'Deleted Mailbox')
- Important: These changes will not be saved until you properly exit the configuration screen. To do this, tap the '
< Account button near the top of the 'Advanced' screen.
- When you get back to the 'Account' screen, tap 'Done' at the top-right corner.
If you'd like a step-by-step guide with screenshots, I have created one and made it available at: http://share.luo.ma/TUAW/guides/Archive-Email-Instead-of-Deleting.pdf