There seems to be a problem with entering birthday data in some
versions of Zarafa Webaccess. It does work in the Webapp.
This is why there is a check box "Send Birthday Information"
This is a Zarafa bug not Tz-push
gcontactsync add its own aniversary field as does tzpush.
You can disable showing the anniversary field in tzpush by setting
tzpush.showanniversary to false in about:config.
tzpush will add in extra fields for phone numbers in Contact main tab.
To hide them set tzpush.hidephones to true in about:config.
You may need this if you use other addons that alter the Contact display.
Set tzpush.debugwbxml to 1 in about:config