I'll take a stab.
Background info: most of us probably thoughr of that Zinc and HCl reaction producing ZnCl2 and H2. It made sense to us back then, it's a redox reaction. Well HCl is a strong acid so it dissociates to H+ and Cl-. The Zn then reduces the H+ and that's how you end up with H2 gas and ZnCl2. Zn has the option of reducing H+ or Cl-, however notice that Chlorine already has a negative charge (it's got enough electrons).
In Cu and HNO3, the same thing happens, we have the acid dissociating into H+ and NO3-, with the Cu ready to reduce. Most people would say "H+" is reduced since it has a positive charge, BUT we need to look at the lewis structure for NO3-.
Notice how nitrogen has a positive charge in all the resonance structures? So now we're trying to decide who wants to be reduced more, N+ or H+? Think about the electronegativities of the two atoms. We KNOW that F, O, and N are more electronegative than H because these atoms can engage in hydrogen bonding. Electronegativity is the tendency is attract electrons to itself. So N+ seems to be the clear winner in terms of who will be getting the electrons. [Electronegativity is 3.04 for N and 2.20 for H]
The reaction with Zn and HCl is easier to deal with because we don't have two electron deficient atoms to choose between. However, what would you expect from Zn and HNO3?