How does the normal force change *while* I'm lifting a heavy object? A heavy weight is resting on a surface with, with the normal force equal to the weight. So the object is stationary. (W = N)
Now I try to lift the heavy object so I'm exerting some upward force U. The object is still stationary, so it must be the case that W = N + U. And that means that the normal force has decreased.
Question: How does the normal force know how much to decrease?
Edit
Trying to clarify the question: 
It's clear to me that Fnet = 0. My question is how does it stay 0? How does the normal force "know" how much it needs to decrease to keep it at 0?
 A: It helps to keep in mind that solid objects, say the floor your heavy weight is resting on, are not perfectly rigid.  They do deform slightly when forces act on them.
Consider resting the heavy weight on a mattress rather than the floor.  You would see the mattress act like a spring and follow some sort of relation where the amount of restoring force it pushes up with increases as the mattress is compressed.
The floor is like the mattress except that it has a much larger spring constant, but it does deform a bit when the heavy weight is resting on it and it deforms a bit less as you start to lift the weight.
A: Fnet=0 draw a FBD and that will show you how it works.
A: According to Newton third law action = reaction in layman's terms, so it's better to rewrite your equation other way around: $\vec{N} = -( \vec{W} + \vec{U})$. Btw, I have extended a little bit U notation - it can be negative or positive as well. If negative - makes normal force smaller and when positive - makes normal force bigger.
