Both Apple and Android phones have the option to turn off mobile data usage. However, this only turns data off for App use. The phone and operating systems (iOS or Android) still have some data requirements that cannot be disabled in the user settings. These are features like 'Find-my-phone' and 'App/Play store' verification.
Some simple/feature phones also have some data services, (and some only in their operating system and firmware), that are used to check for updates and network registration.
If you wish to disable mobile data completely then you can do so in your 1pMobile account area in the Network Settings. However, this may prevent the operating system's features from working correctly.