If any of their support tries to tell you all the standard **** about clearing your browser cache, logging out etc. it's just their way of ignoring this issue for a while longer while making you go through all these meaningless procedures that imply the issue is isolated to you and your particular browser or environment. In other words the problem is yours not theirs, when the evidence is clear, they have a bug on their end and the problem is theirs not ours.
I created a brand new email account for our organization yesterday. The new email was for a new employee. The email account was accessed for the first time from a new computer and browser that had no previous activity of any kind, least of all any Microsoft logins. By the end of the first day they started experiencing the same behavior of being logged out for no reason.
This is a Microsoft problem and given the crickets we've been hearing from them on this issue for nearly a month now, it's pretty clear they don't actually care enough to prioritize this.
I'm going to make sure this thread is not dropped and continues to grow by posting a daily update until someone notices us lowly peons and helps... or until next Friday when we finally decide to move our company back to Google where we should have stayed to begin with.