The question on whether virtual particles (gravitons) can interfere is preceded by some arguments that contain a number of statements in need of careful consideration. So before I address the question, I'll first look at these statements:
"Their existence [virtual particles] is allowed by Heisenberg's
uncertainty principle, which allows a transitory violation of energy
conservation in processes that endure a short enough time."
This statement reveals a common misconception about Heisenberg's uncertainty principle. It actually specifies a minimum uncertainty and not a maximum uncertainty. In other words, to satisfy the Heisenberg uncertainty principle, something needs to exist a long enough time and not a short enough time. Something cannot `hide underneath' the uncertainty. No, the principle states that it is fundamentally impossible for anything (even virtual particles) to exist for shorter than a certain time given by the energy uncertainty.
"...it is established that the less energetic the force carrier, the
stronger is the force mediated."
Actually this is only true for non-Abelian theories (such as QCD and electroweak), as far as I know. Their couplings become weaker for higher energies. Other theories, such as QED grow stranger at higher energies. A quick survey of literature seems to indicate that there is a disagreement on whether quantum gravity can be said to have a running coupling.
If gravitons can interfere, then it is likely that they would
interfere destructively (chance dictates that both constructive and
destructive interference are present, but destructive interference is
more likely, that we can prove)
I would like to see that proof. Perhaps you can give a reference. Heuristically one can argue that destructive and constructive interference should be equally likely. The reason is that there must be energy conservation. In a random light field for instance one would find speckle, which contain bright spots and dark spots, such that the average intensity would maintain energy conservation.
From Planck's relation from quantum mechanics, that means that these
gravitons will be less energetic, thus increasing the strength of the
force of gravitation, over large scales.
Based on the concerns mentioned above, this conclusion is questionable. In fact it is basically ruled out by observations. A force that grows stronger with distance would arguably reach a distance beyond which it become non-perturbative. It other words, it would confine itself just like QCD does. We see absolutely no evidence for that.
So let's consider the question about whether virtual particles can interfere. (In the context in which this question is being asked, I seriously doubt that this answer, or any of the other answers, would address the actually issue that the OP wishes to raise. There are simply too many misconceptions. However, for the sake of the question itself, and because the current two answers disagree with each other, I'll throw in my attempt.)
Virtual fields are perhaps most rigorously found as being represented by the internal lines in Feynman. The reason for them being virtual is because they are off-shell - i.e. they do not satisfy any dispersion relations. From this understanding it then followed that some physicists suggested that the force fields around particles (such as the Coulomb field around a charged particle) can be viewed as a virtual field.
When we want to say whether these virtual fields can interfere, we need to agree about what we mean by interference. It most situations this is understood to be found when different fields can form a linear superposition. Under certain situations this superposition may be observed as interference fringes. However, fringes do not need to be present to represent an interference phenomenon. There is for instance also the case of quantum interference in, for instance the Hong-Ou-Mandel effect.
So if we restrict ourselves to the case of virtual fields as they appear in Feynman diagram. Then indeed we have a linear superposition of infinitely many different contractions of the different fields (plane waves, both on and off shell). This linear superposition exists in the form of integrals over all the phase space degree of freedom that are involved in the diagram. Well that would mean that we have the scenario of interference, and since in incorporates virtual field, it then follows that such virtual fields also interfere.
Sadly, such an interference cannot be observed in the same way that one would observe interference in a classical optical field. The effect of the interference is more a sum-over-histories kind of interference that determined the probability amplitude for a specific interaction.
In view of the above, it is strongly doubtful that this could explain dark matter.