The problem is that if the submitted email doesn't match the user's current email, it tries to reset the email field. If you removed the email input fields completely then it's trying to set a blank email, which causes that message.
You could solve it by changing the email and emailconfirm fields to hidden fields instead of deleting them, but it might be better to create a plugin to stop email changes (because just removing or hiding the fields might not stop someone from changing their email if they know that's all you did). The plugin would use hook profile_updatepassword_start and have code like this:
Code:
$vbulletin->GPC['email'] = $vbulletin->userinfo['email'];
$vbulletin->GPC['emailconfirm'] = $vbulletin->userinfo['email'];