BirdieSync synchronizes contact emails if their associated label is "Home" or "Work" on the iPhone. "Home email" is mapped to "Email" in Thunderbird and "Work email" is mapped to "Additional email" in Thunderbird. So if for instance the email is associated to "Other" label on the iPhone, it is not synchronized.
If you chose to synchronize your contacts on your iPhone with an Exchange server, then your contact emails are no longer associated with a label on your iPhone. You cannot select a specific label if you create a new contact on your iPhone or edit it. So in this case, emails cannot be synchronized by BirdieSync (emails will remain blank in Thunderbird). In order to be able to choose the label of your emails on your iPhone and synchronize them with BirdieSync, you must open Settings/Mails, Contacts, Calendars/Contacts/Default account" and select "On My iPhone"). Which also means that your new contacts will not be synchronized with your Exchange server.
Note that if you use Lotus Notes Traveler on your iPhone, you use an Exchange server to synchronize your contacts.