The magnet on my cell phone case cuts my phone off, will this damage my phone?

Magnets and electronics do not mix, EVER there is bound to be some metal that is attracted in your phone and its not a good idea, it can also wipe memory

A lot of phone cases now have magnets in them, and it shouldn't damage the LCD screen like it would your old television. If it's causing the phone to turn off when it's near a magnet, it may be disrupting some of the contacts internally. In that case, it could damage it. Get a body glove style case that doesn't have a magnet.

You may have more of an issue with the magnet affecting the signal of the phone thus throwing it into a Searching for signal mode... Thus draining your battery.