At this time Office365 does not post emails sent from outside of their system (even if using their SMTP server settings to the sent folder like other clients do, such as Gmail). PhoneBurner has contacted Microsoft directly and they have been clear this is not something they do.
A possible workaround is to use the bcc option to get a copy of the email sent to their Office365 account. To do so, follow these instructions:
1. Click the account icon in the top right hand corner of your PhoneBurner account
2. Click Other|Back Office Settings
3. Fill out the information for Auto BCC All Email and click Save Changes