Just spoke with Google about this...
It is a known Google Sync issue with iOS, see here:
http://support.google.com/mobile/bin/answer.py?hl=en&answer=139635
The symptoms are the same for all iOS devices running Google Mail set up as Exchange (apparently this just applies to Google Apps / hosted mail, as GMail folders sync fine unless manually set up as Exchange).
iOS automatically saves Exchange drafts into a local-only Drafts folder (the one with the note/paper icon next to it). If you scroll down you'll see another Drafts folder which is Google's server Drafts folder. You will be unable to change the settings as of this writing to force the phone to save into the server Drafts.
You will also be unable to pull up a locally-saved draft message on the device and re-file it into another folder (namely, the server Drafts). The only workaround I've found is to re-address the draft message to myself, send it, and then call it up on a computer from my Inbox and save it to the Server Drafts.
If anyone figures out a better way, please post!
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Sorry. I just tried creating a draft in my Google Mail using the web interface on my MBA, and after I saved it, it appeared on my iPad. I don't see an option where you can turn that on or off. It just worked.
This way works as well as a workaround, but not if you create the draft on the mobile device. Another workaround for doing it that way would be to use the
web interface on the mobile device, and not the iOS Mail app when you draft your message initially.