Important note: The answer below was true for Core NFC in iOS 11. A lot has happened since then. In 2019, Apple added many enhancements to Core NFC. The API now provides access to lower protocol layers (such as exchanging APDUs with contactless smartcards (ISO/IEC 14443-4), and sending commands to MIFARE (and potentially, but not verified, other ISO 14443-3A tags), FeliCa, and ISO/IEC 15693 tags).
The Core NFC overview page announces that as:
Your app can also write data to tags, and interact with protocol specific tag such as ISO 7816, ISO 15693, FeliCa™, and MIFARE® tags.
Answer from 2017:
No, Core NFC only works with NFC tag that are NDEF formatted. Thus, you can only interact with NFC tags that adhere to the NDEF (NFC Data Exchange Format) hardware abstraction layer specified by the NFC Forum. Specifically, with NFC Forum tag types 1 to 5.
Core NFC overview page:
Reading NFC NDEF tags is supported on iPhone 7 and iPhone 7 Plus.
Using Core NFC, you can read Near Field Communication (NFC) tags of
types 1 through 5 that contain data in the NFC Data Exchange Format
(NDEF).
Contactless smartcards (like electronic passports, payment cards, etc.) and even additional protection features of NFC tags (like NTAG password protection, MIFARE DESFire or Ultralight authentication, etc.) cannot be accessed using the iOS 11 NFC API.