The delete button is grayed out on communications that were sent via a correspondence process, which means that only records that were added manually to the communications tab can be removed with the Delete button. However, there is a way to remove it from the record on the back-end.
You'll need to identify the constituent record's system record ID (GUID) from the Constituent table.
Next you'll need to find the communication you'd like to delete in the following tables: INVITEE and INVITEEHISTORY.
Both the INVITEE and INVITEEHISTORY tables can be searched using the CONSTITUENTID, which should help you find the entry in question.
Delete the communication from both tables and you'll find that the entry has been removed from the communications tab of the constituent record.