Using a phone during Tavaf is permissible when the phone is being used as a tool to aid the worship — such as reading Kur'an from a Kur'an application, accessing a dua collection, or using a counter app to track the circuits of Tavaf. These are modern tools that serve the same purpose as a printed dua book, which scholars have always permitted during Tavaf. Reading from a book during Tavaf was practiced by some of the Salaf.
Sheikh Ibn Uthaymeen, while he did not address smartphones specifically, stated that reading duas from a book during Tavaf is permissible. By extension, contemporary scholars permit using a phone application for the same purpose. The key is that the phone is being used for worship-related purposes, not for worldly distractions.
However, scholars unanimously discourage non-worship phone use during Tavaf: taking or making phone calls, sending messages, browsing social media, or taking selfies. These activities distract the haci from the remembrance of Allah, which is the primary purpose of Tavaf. The Permanent Committee has advised hacilar to minimize distractions during the rites and to focus their hearts and minds on worship. If a phone call is genuinely urgent, the haci may step aside, as breaks during Tavaf are permissible, take the call, and then resume their Tavaf from where they stopped.