Here's the reality, plain and simple:
Electronic technology is inherently sensitive to high-strength magnetic fields. Small magnets, however, will not cause any sort of drastic damage to your phone as the posts of this thread express concern over. It's
possible that very high-strength fields could damage your phone's main board, but it is highly unlikely -- and you are very unlikely to come in contact with magnetic fields of that magnitude. NFC technology does not in any way rely on magnetically stored information (such as that of a credit/debit card), because NFC is simply a close-proximity radio frequency data exchange standard. Though strong magnets may produce interference to active NFC exchanges, the type of magnets used in phone accessories will not cause any such interference. In fact, you can purchase magnetic NFC stickers/tags online; they'll work perfectly.
Most modern smartphones use a magnetometer to determine which cardinal direction your phone is facing. Magnetometers act very much like a traditional compass; they sense the magnetic fields present at your location in order to determine the direction of magnetic north. Even though the magnet on your phone's case is very weak compared to the magnetic field of our planet, having it right next to the magnetometer results in a much stronger perceived strength. Try moving your case's magnet around a traditional compass -- it's the same as what's happening with your phone.
So no, it's not gonna kill your phone.