I thought about storing important and sensitive information in iOS' keychain. But now I read that the keychain is only restored if the backup is encrypted in iTunes (don't know about iCloud backups). This is especially a problem when users buy a new iPhone/iPad and restore them from a backup. The information stored in the keychain by the old device will be lost.
Is there any (secure) possibility to transfer the data to new devices or on restores independently of the backup settings?
You can transfer data using the iCloud, but this is NOT really secure! Save this data ENCRYPTED on your server and let the app read this data would be a solution. Btw. you shouldn't save sensible data unencrypted in the keychain. The keychain can be read really easy after jailbreak.
For more information about handling with sensible data you may read this book: Hacking and securing iOS Applications