It seems that there are users who are encountering an issue when trying to send emails using either service and are greeted by a “Cannot Send Mail” error message. The good news is that it appears that Apple is aware of the problem and are working together with Microsoft to come up with a fix for it.
A post on Apple’s support page reads, “If your email account is hosted by Microsoft on Outlook.com or Office 365, or an Exchange Server 2016 running on Windows Server 2016, you might see this error message when you try to send an email with iOS 11: ‘Cannot Send Mail. The message was rejected by the server.’ Apple is working closely with Microsoft to resolve the issue and will release a fix soon in an upcoming software update.”
In the meantime for users who absolutely have to use these services, you can try logging in through a browser like Safari to send your emails, or alternatively you can try downloading the Outlook app from the iTunes App Store that should circumvent the problem.