How are you sending emails? Built-in PHP option or SMTP? If it's SMTP, then nothing should have changed as it'd still be using the same server (unless your new host silently limits this).
If it's PHP mail, make sure you have the -f parameter enabled. If you have this enabled, then your new host likely doesn't allow it to be used by the user that PHP is running as. You can ask them about this but it's unlikely that you can override it.
Note that you can confirm if it's working correctly by looking at the "Return-Path" header of an email generated by XenForo.