Initial Condition in a Numerical Problem

In a initial value problem does the initial condition has to satisfy the boundary condition and the governing equation?

For example: If a non-homogeneous Neumann boundary condition for the pressure in a fluid dynamics problem is applied on a boundary, can one initialize a zero pressure field?

That depends on the equation you have, and on the situation you want to model. Imagine, for example, that you are considering the advection equation $$\partial_t u + c \partial_x u = 0,$$ i.e., you are transporting a concentration of a substance along with velocity $c$ from the left to the right. It does make sense to say that at time $t=0$, the concentration everywhere is zero, i.e. $u(x,0)=0$ but that the "reservoir" that lies beyond the left edge has a concentration 1, i.e. $u(0,t)=1$, so that a fluid with a high concentration flows in from the left, replacing the fluid with low concentration that was there before.
Likewise, if you consider the heat equation $$\partial_t u - k \partial_{xx} u = 0,$$ then you can think of a situation where you model a solid that has initial temperature $u(x,0)=0$ but that starting at time $t=0$, you put a heat bath with temperature 1 in contact with the left or right end of the solid, i.e., $u(0,t)=1$, so that it heats the solid from the left.
On the other hand, if you had the Stokes equation $$\partial_t u - \eta \Delta u + \nabla p= 0, \\ \nabla\cdot u = 0,$$ then you also have to prescribe initial and boundary values for the velocity. Here, I cannot immediately conjure a situation where the initial velocity is zero (which implies that up until $t=0$, the walls that enclose the fluid are also at rest), but that starting at $t=0$ the walls move at a certain velocity, thereby imparting a nonzero boundary value on the velocity. That is because you would have to accelerate the walls from rest to a nonzero velocity instantaneously -- something that require infinite acceleration and consequently infinite forces.