What does this ribbon on the iPhone 5s LCD shield plate do? While opening up my iPhone 5s to replace the screen, I ripped this ribbon that was stuck to the LCD shield plate and I was wondering what it's called and what it's function is?
It's the grounding plane.  Basically it reduces electrical noise and interference. Replacing it would be ideal but, if it's a very small rip, you should be okay.  Very small being 1mm or less.  It may function with a bigger rip but I'm not sure how that will affect the performance of your device.