Why does Lift and Drag calculation over an airfoil ignore shear stress?
We "Do" take viscous shear stresses into account when calculating the drag of an airfoil.
This is ALWAYS true for subsonic flows. The underlying reason behind this is called D'Alembert's paradox. For incompressible and inviscid potential flow, the drag force is zero on a body moving with constant velocity relative to the fluid. 

BOTH pressure drag and profile drag on an airfoil is viscous phenomena. 

In order to provide a numerical example, please see the inviscid and viscous calculations of SD7037 airfoil. What you see in the inviscid case as drag, is merely a numerical error (- value).
 

PS: In order to really understand how drag is formed, best to start looking up the "Von Kármán Momentum Integral" equation. Also to convince on the analytical results one can use XFOIL's VPlo menu under OPER and observe the momentum defect via delta* distribution and kinetic energy defect via CD distribution