If I try to go to the settings in System Prefs > MobileMe, it asks for a username and password. Is there any way to remove the menu bar icon without needing to either login or create a MobileMe account?
If you mean this icon:
You must run iSync (by QuickSilver or Spotlight خق go into /Applications and double-click iSync) then uncheck "Show status in menu bar".
You can remove menu icons by dragging them out of the menubar while holding the ⌘-key.
This behavior does not work with icons from third party applications (e.g. Dropbox, Growl...).
Apart from dragging it out as already mentioned, MobileMe specifically can also be set to show via the MobileMe System Preferences. Just untick the box stating Show MobileMe Sync Status in menu bar. This obviously only works if you have a MobileMe account.