Depends on what you mean. Shared mailbox on its own cannot send messages, as there are no credentials (no way to access it directly). Granting Send As permissions allows delegates to send messages using the email@example.com address though.
firstname.lastname@example.org is a normal mailbox before converted to shared mailbox. Application is using this address to send mail out from their applications. After converted email@example.com to shared mailbox, the recipients are not receiving the mails. Correct me that this is the correct behavior ? If so, is there a work around?