To disable your caller ID on iphone is very simple, all you will have to do is to follow the steps below.
1. Go to your settings on iphone, Open it and scroll down and click on phone in the list.
2.Then select show "my caller ID" here you will be able to turn it on or off. so to disable it, simply turn it off
NOTE: This feature is not available in all countries, Therefore you may not be able to use this method.