Nexus 4 Screen Replacement - Would missing shielding tape or double-sided tape cause issues?
The chip that's missing the shielding is part of the digitizer; it is responsible for interpreting changes in the electric field of the screen and figuring out where your finger currently is, and what location of the display you've touched. Without shielding it won't work properly, or as you've seen.. at all. Get some here.. any decent RF shielding tape ought to work.

The conductive foam stuff is called an EMI gasket, and is used to isolate RF noise sources by "trapping" noise sources in specific regions. It more or less works the same as a traditional gasket, physically blocking the path. Measure the old piece if you can, or just guesstimate the size required.