Will wearing multiple fire protection armor pieces stack their effects?
Yes, fire protection stacks. To test this, I put on a diamond helmet with Fire Protection I and stood in fire. Then, I put on an entire suit of diamond armor with Fire Protection I and went to stand in the fire again. I took no damage, though my armor took extensive damage.
This is the video I made in the process (I'm not sure I've properly annotated it, but hopefully you can tell what I'm doing. :P):
They definitely stack, I just tested them. You're pretty much immune to fire with two diamond pieces of fire IV if your hunger bar is full.