A book sits on a table. What is the net force of air pressure? An elementary problem asks,

A book is at rest on a table top. In what direction is the net force
  of air pressure on the book?

Is this a meaningful question, and if so, what's the answer?
If we imagine that the book and table are completely smooth, there's no air between the book and table. Then the air pressure from above the book creates a large net downward force of hundreds of pounds. (This will be countered by an upward force from the table.)
On the other hand, if we imagine that due to the rough nature of the book and table, there is some air between the book and the table at most places, maybe there is enough air underneath the book to provide a net pressure force that is upwards. The scale height of the atmosphere is about 10^4m, so a 1cm book needs to have less than 1 part in 10^6 in contact with the table to have net upward force from air pressure.
How realistic are these approaches? Do we need a molecular view of the book, table, and air to understand the situation? For a typical, everyday book and table, is it meaningful to ask what direction the net force of air pressure pushes?
 A: 
How realistic are these approaches? Do we need a molecular view of the book, table, and air to understand the situation?

It is realistic. There are no need to consider the molecular nature of air.
See http://www.nanovea.com/Application%20Notes/paperroughness.pdf. According to their data, the typical length in paper's surface variation is in order of $10^{-5}$ meters, while air's mean free path is $6.8\times 10^{-8}$ m (at room temperature, ambient pressure.) That means the room below the book cannot be thin enough to prevent significant amount of air sitting there (Babou actually reasoned his/her answer from wrong assumptions)
Considering the deformation of 'peaks' of surface of cover of a (heavy) book, it is likely that the book is in contact with the table more than 1 part in $10^6$. Therefore I'd guess that the net force is downward.
However, our static approach cannot be used if someone is picking up the book. S/he may experience the 'suction cup' effect (described in Babou's answer) when the book s/he lifted feels like "glued" to the table at one instant. When the book is lifted, air trapped below the book experiences a rapid, adiabatic expansion. Viscosity prevents surrounding air from entering the expanding room (below the book's surface) so rapidly. Hence the pressure below the book drops, and pressure from air above the book wins.
A: We will assume that the book and the air
layer between book and table are thin enough, possibly no air in places, and the book dense
enough (very important), so that atmospheric pressure may be
considered constant, i.e., so that its variation can be neglected.
In a nutshell, the answer is that there is a downward force from atmospheric pressure. It can be null. The reason is that the pressure on the sides balances. The pressure
downward applies to all of the top of the book.  The pressure upward may apply to only part of the bottom surface of the book. This can be seen with a book that has a rubber cover and is placed on a glasstop.
Essentially the you have a suction cup effect that may make it very difficult to lift the book, much more than its weight would warrant. It can actually be measured by a dynamometre. So the net effect of atmospheric pressure is downward.
Actually it can be upward if you use atmospheric pressure to "glue" your book to the underside of a glass tabletop (better use one with a rubber cover).
It is the same phenomenon that happens when you need extreme force to pull the plug
from a full bathtub.  And it is also what has drowned a few people in
swimming pools.  They could not detach themselves from an open water
exit at the bottom.
Note that you can also have trapped bubbles of compressed air that have the opposite effect.
I had made a complete model of the system (as you started discussing details), but I realized that was not what you asked.
A: Is the book and table one object or two?
Either way, the large downward force on the top surface is matched by an equal force on the bottom surface.
If it weren't, the object(s) would experience acceleration.
