Recently I experienced an issue with utilising outlook.office365.com to access Office 365 mail from Outlook, Web (OWA) and iOS apps when on a particular internet connection. This only affected Apple devices and I had no issues with using Windows based computers on the affected network.
Interestingly I have two internet services with this particular ISP and I had no issues accessing outlook.office365.com on the other service. When the issue was experienced, I connected to my 3/4G connection and issue solved.
After some extensive troubleshooting of the router, I tested changing the default DNS address (which was 192.168.1.1) to one of the DNS addresses belonging to my ISP. Once this change was applied, I was then able to connect to Outlook, OWA and iOS apps.
I am still not certain of what is causing this issue, but at least I have identified a workaround to get me back online.