I am trying to get the face to show as transparent through the mug object which is a live paint group. I've tried the Pathfinder, but it didn't work. What do I need to do?
You can't use Pathfinder on a Live Paint group. You need to expand the group first (Object → Live Paint → Expand or just hit the "Expand" button in the top bar).
Pathfinder doesn't work with strokes either so you will need to outline your strokes if the white objects are made with strokes (Object → Path → Outline Stroke).
The Pathfinder "Minus Front" function doesn't work well with groups either so you will either have to cut out each object individually, create a compound path out of all the white objects (Object → Compound Path → Make or cmd+8) or you can use Pathfinder's "Divide" and manually delete the bits you don't need.
A better (non-destructive) solution is to use a transparency knockout group, as described here: