I have an unattended mailbox that sends emails to some customers.

sometimes the email is not in the sytem correctly, and you get a NDR
message.

I have created a rule in the mailbox to forward the NDRs to a certain
user, but was wondering if we have to open outlook for that unattended
mailbox to process its rules?

Maybe.

If the rules are server based, then no.

If the rules are client based, then yes.

When you create a rule, if it says 'This rule is a client-only rule, and
will process only when Outlook is running.' then it is a client-only rule.
Also, client-only rules say '(client-only)' in the list of rules.

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f.h.
Microsoft Outlook MVP