Wormhole traversability in general relativity It is well known that stable, traversable wormholes require exotic matter that violate the dominant energy conditions.
For the purpose of the question, let's assume enough exotic matter can be created for a brief period of time, where the wormhole mouths are stable and traversable. Additionally, suppose that during that time, a strong beam or wire is introduced across the throats. Now the question is:

Question: What happens if exotic matter is removed from the wormhole mouths? Are the tidal forces always enough to break any
  material crossing the throats and disconnect the mouths?

 A: If the exotic matter were suddenly removed from a stable wormhole that is spherically symmetrical and nonrotating, it would become a black hole.  This immediately follows from Birkhoff's Theorem -- the only spherically symmetrical solution to the Einstein field equations in matter-free space is the Schwarzschild (non-rotating Black hole) solution.
Observers on both sides of the wormhole would note the disappearance behind an event horizon of that part of the wire nearest to the wormhole's throat.  
To observers stationary relative to the newly formed black hole, the black hole would appear frozen (as is well known) with the wire jutting out of the event horizon.
To observers suitably accelerating toward the black hole, the black hole would appear to pinch off (as is well known) with the wire being severed.
The only proviso in this explanation is that the presence of the wire makes spherical symmetry only approximate.   I have assumed, perhaps incorrectly, that the solution perturbed by the presence of the wire is essentially the same as that in the wire's absence.
Sources:  Gravitation by C. Misner, K. Thorne, & J. Wheeler (1973), The Physics of Stargates --Parallel Universes, Time Travel and the Enigma of Wormhole Physics by E. Rodrigo (2010)
Minor quibble: Matter's violation of the Dominant Energy Condition is not sufficient  to support a stable wormhole.  For such a wormhole to allow the traversal of material objects, its matter must violate the Weak Energy Condition.  [In other words, it's possible to have matter that violates the DEC that is insufficiently exotic to support a wormhole.]
A: As of 29 October 2013, we now have a wormhole metric that can be generated by a non-exotic stress energy tensor. The research paper ( which isn't really too dense ) Passing The Einstein-Rosen Bridge goes through the GR calculations of timelike geodesics through the Einstein-Rosen Bridge(wormhole) that do not terminate in a singularity in a wormhole. Instead, they have managed to shift the singularity out of the timelike region of the spacetime into the spacelike region: a place where our geodesics can't pass to.
The singularity being in a timelike region of spacetime is what required the existense of exotic matter, because we would have to create some kind of "shielding" from our worldline terminating at the singularity. Doing so requires exotic matter, unfortunately.
Now that we have a decent wormhole metric that's a function of a non exotic stress-energy tensor, it's just a matter of working out the finer details, and it's up to our engineers to get us physically testing this.
A: This is tentative answer and not final because I have no proof for assertion I:

Assertion I: when exotic matter is removed from a traversable wormhole, and regardless of what normal matter distribution already exists inside and outside the throats, a surface inside the tunnel that is orthogonal to the
  longitudinal axis is formed such that all light cones from both throats become
  asymptotically closer to the surface without never touching it.

Assuming Assertion I is true, then two elements of the beam will have to be on opposite sides of the dividing surface. Since they are on opposite sides of the dividing surface, they will not be able to communicate to each other, which means that the speed of sound on the material will have to exceed the speed of light in vacuum. This proves that no physically realisable material will be able to support the wormhole open without exotic matter
