Why can we assume long structural members to be in plane strain? 
In plane strain, the strain in one direction is assumed to be zero. It is taught that if we have a structural member that is very long in one direction compared to the others, the strain in the long direction can be assumed to be zero. A dam is a popular example is textbooks. But why is this? I've never seen this explained very well. It is simply stated that the strain (not close to the ends) is assumed to be plane strain. Why can we assume this? What does the length of the dam have to with it?
And if this is true, why don't beams follow this same principle? If we have a long beam, we should be able to assume the normal strain along the length of the beam is zero. But according to Euler-Bernoulli beam theory, this is not the case. In fact, in E-B theory we assume to normal stress along the length of the beam is the only non-zero strain. 
 A: 
It is taught that if we have a structural member that is very long in
  one direction compared to the others, the strain in the long direction
  can be assumed to be zero.

You are misunderstanding what the book is trying to explain.
The strain in the long direction is zero because of the boundary conditions at the two ends of the structure, not because the object is "long".
For example the ends of a dam are built into the sides of the river valley and cannot move apart. Therefore the axial strain must be zero. 
The same thing is true for a long pressurized pipe. The pipe can't move axially relative to the ground, so the axial strain must be zero.
For a cantilever beam, one end is free and there is nothing to prevent it moving axially.
Actually the dam is only a good example if it is a straight dam. Most real-world large dams are curved to form an arch turned on its side, and in that case the dam can have axial strain if it changes its curvature slightly, even if both ends are fixed.
A: I think that we can often deal with long structural members as plain strain due to friction along the length, which acts as a practical restriction to displacement.
In the case of the dam, the friction with the soil under its own huge weight avoids longitudinal displacement.
Another example is rolling of steel sheets. The roll gap, being thinner than the incoming sheet, forces a reduction of thickness. The material can flow forward much easily than sideways, due to friction with the rolls. (by the way, the process requires friction, otherwise the rolls can not "bite" the entry bar). The result is an elongated sheet with almost the same width. This is plastic deformation not elastic strain, but the reason for the outcome is the plane strain configuration.
In the case of a steel beam there is no restriction to longitudinal displacement. 
