My student, Brad Rodgers, has just posted a paper on the arXiv at http://arxiv.org/abs/1203.3275 which proves a partial result towards the repulsion effect (that differences of two imaginary parts of Riemann zeroes tend to avoid another Riemann zero), in the spirit of Montgomery's partial result towards his pair correlation conjecture (i.e. this repulsion effect can be detected when tested against sufficiently band-limited test functions, assuming RH).
Ultimately, the reason for this repulsion lies in the obvious approximate formula
$$ |\Lambda(n)|^2 \approx \Lambda(n) \log n$$
where $\Lambda$ is the von Mangoldt function. If one compares this with the explicit formula, which is formally of the form
$$ \Lambda(n) = 1 - \sum_\rho n^{\rho-1} + \ldots$$
one begins to see the negative correlation between differences of imaginary parts of zeroes $\rho$ (which show up in the expansion of $|\Lambda|^2$) and in the imaginary parts of zeroes themselves. (Making this intuition rigorous, though, is somewhat non-trivial, requiring manipulations similar to those in Montgomery's original paper to deal with the fact that the explicit formula as given above is only convergent in a very weak sense.)
EDIT: it is likely that a similar analysis would also explain why Riemann zeroes correlate with differences of zeroes of other functions. For instance, starting from $|\Lambda(n) \chi(n)|^2 \approx \Lambda(n) \log n$, one can predict that differences of imaginary parts of zeroes of a Dirichlet L-function should repel away from the imaginary part of zeroes of zeta.
