Recently, I had a case with an Exchange Online customer where they didn't feel an automated reply on a mailbox was working. While digging a little deeper I found out that the behavior they were feeling wasn't correct was the following scenario:
Group mailbox had an automated reply set. It is important to note that while the reply was automated, it was created using the rules wizard, not the OOF wizard.
If e-mail is sent directly to the mailbox, the sender receives the reply. If however an e-mail is sent to a distribution group the mailbox is a member of, the reply never shows up. We have documented this scenario in 2007 and 2010 that the fix for this is to check the box for Allow Out Of Office replies to sender in the properties of the Distribution Group. However, we do not expose this attribute in the GUI in 2013. Instead the following powershell line needs to be run:
Set-DistributionGroup aliasofDG –SendOofMessageToOriginatorEnabled $True
Once that is set, the rule fired properly and the reply is received by the sender. The down side to this of course is if anyone else on the DG sets OOF, then the sender will also get their OOF message.