Can the Casimir effect also generate a repulsive force at large distances? I've just seen a documentary where they state that particles pop into existence and then annihilate each other all the time in empty space. The experiment discussed that proved this was that two metal plates placed very, very near each other would be pushed together. 
It occurs to me that if you placed them very, very far apart, such that more of the universe was between them than was outside them, wouldn't they be pushed apart for the same reasons? Could this be what causes the universe to expand? It seems to me that if there's any pushing at all, there has to be expansion.
 A: This is the experiment you cited:  Two uncharged parallel plates separated by a tiny distance, on the order of the wavelength of a single photon, will limit the photons popping in and out of existence between them to those whose wavelengths can fit a whole number of times between the plates.
As the plates draw closer together, one should find that the photons popping in and out of existence between them consist only of those with such whole wavelengths, and should exclude photons of all other wavelengths.  This should reduce the total vacuum energy between the plates, as compared to the vacuum energy in space around the plates, which implies that the plates are drawn together by the reduced vacuum energy.  The attraction is called the Casimir effect, as Alpha001 notes in his comment.
Another way of looking at the effect is as though the two metal plates are connected by a stretched spring (a sea of virtual photons), and move together as the stretched spring loses energy (limitation of the virtual photons) and contracts.  This metaphor was devised by Stephen Reucroft and John Swain.
Moving up in scale to the entire cosmos, quantum field theory views space to be filled with virtual energy that fluctuates in and out of existence.  But unless there are boundaries that limit the virtual energy to certain whole wavelengths in a bounded area, there will be no Casimir effect.
A problem with your interesting suggestion is that it is not believed that the Universe is finite in the sense of having a boundary, and is not divisible into one area containing "more" Universe, and another containing "less" Universe, which sum to the entire Universe, as Countto10 points out in his comment.
Another problem is that even if there were evidence that the Universe were finite and divisible into two parts that sum to the whole, General Relativity provides that the difference in vacuum energy between the two parts of the Universe might cause recognizable curvature, which has not been found.
