I am using iOS 8.3 with VoiceOver.
1. Open the dialog for a Contact which has several alternate e-mail addresses.
2. Double tap the Edit button.
3. Flick to one of the e-mail addresses for this contact that you want to delete.
4. VoiceOver indicates that you can swipe up/down and choose and action.
5. Swipe up/down until you come to "Delete".
6. Double tap "Delete".
7. Note: Nothing happens. Additionally, the "Done" button in this dialog is still grayed out indicating that no changes have been made. Thus, it is not possible to delete this e-mail entry.
8. Additionally, if one flicks right, one will come to a Delete button after being on the e-mail address. Unfortunately, it is not easy to determine whether this button applies to the previous e-mail address or the next e-mail address when there are multiple e-mail addresses for a particular contact. In any case, double tapping this Delete button also has no effect and the "Done" button in the dialog remains grayed out.
Summary: It is impossible to delete one of multiple e-mail addresses for a particular contact. In addition, it is not clear which e-mail address the Delete button refers to when there are multiple e-mail addresses for the contact.