iPhone 12 Pro motherboard replacement and issues that come with it?

It's worth noting that a battery replacement may resolve the boot loop problems. I recommend replacing that before proceeding to a logic board replacement as a more economical solution. Use a known-good battery to test if needed. You can also attempt a smaller type of repair on the logic board itself through micro-soldering. Most components are extremely cheap (pennies) to source.

To directly answer your question:

• Data not backed up to iCloud or an external computer will be lost.
• FaceID functionality will be lost (unless you have a paired logic board + front camera). This means functions that would ordinarily be completed with FaceID will always require a passcode to be entered.
• "Check Engine" message for the battery and display will be apparent, notifying the user that the logic board can't validate the status of the battery or the display because they are no longer paired via serialization. This practically doesn't mean much, but it can be annoying.