If the screen of an iPhone 13 is replaced by a third party that has no official authority from Apple, Face ID will no longer work. That is the conclusion of YouTuber Phone Repair Guru after his own experiments.
In two videos, Phone Repair Guru shows that Face ID stops working after changing the screen. The YouTuber uses original Apple screens, from two copies of the iPhone 13 Pro. He shows that the microphone, proximity sensor and ambient light sensor have no influence on the functioning of Face ID. The module with those components, which is glued to the screen, can be exchanged.
The Face ID components themselves are not part of the screen, but the screen module does contain chips that are presumably linked to the front camera. According to the repairer, it would be possible to swap those chips and that should allow for an unofficial screen swap, but it is unlikely that independent repairers would perform such an operation when replacing a screen.
Apple is known for making repairs difficult when they are performed by parties that are not affiliated with Apple. With previous iPhones, users have already been notified that a non-original screen is being used if the screen has been replaced by an independent party; even if it is an original screen. However, Face ID continued to work on previous iPhones.