If an Office 365 Distribution Group is being synced from on-premise Active Directory environment then you cannot receive emails to that group from an external source by default. If you try to change the setting in Office 365 the following error appears:
The action ‘Set-DistributionGroup’, ‘RequireSenderAuthenticationEnabled’, can’t be performed on the object ‘[Your Distribution Group]’ because the object is being synchronized from your on-premises organization. This action should be performed on the object in your on-premises organization.
This can be changed using the ADSIEdit tool. Navigate through the Active Directory hierarchy and find the Distribution Group then right click and choose Properties. Scroll down to “msExchRequireAuthToSendTo” and change the setting to FALSE. Run a manual DirSync to effect the changes in Office 365.