How can I block my caller ID in my iOS (iPhone) app?

As an iPhone user, you don't need to activate the caller ID blocking in the class settings.

Go directly to the emergency contact in "class".
Click on the phone symbol next to the telephone number you want to call. 
Choose "Copy phone number".
Go to the numeric keypad of your phone and paste in the number.
The prefix #31# will automatically be added to block your caller ID.