The expansion of the universe won't necessarily get to the point where it rips everything apart. It depends on the equation of state of the dark energy. The phenomenon you describe is called the Big Rip, but if dark energy behaves like a cosmological constant (which is currently consistent with observations) there will be no Big Rip.
But the expansion caused by dark energy does form the equivalent of an event horizon. At some distance from it the expansion is so great that nothing travelling at the speed of light or less could reach us even given infinite time. This causes a horizon similar to the horizon round a black hole, and just like a black hole this horizon does emit Hawking radiation. The Wikipedia article on dark energy discusses this, and places the horizon at 16 billion light years so it will be a while before we see it :-)
The paper Expanding Confusion goes into the maths if you fancy the challenge.
Response to comment:
The virtual particle pair being pulled apart description of Hawking radiation is only a metaphor. Remember that the Feynmann diagrams showing virtual particles are just an illustration of one term in a calculation and are not intended to be taken literally. Depending on who you ask, and what you mean by exist, virtual particles don't really exist. See this article by Matt Strassler on virtual particles for an excellent, though inevitably difficult, explanation of virtual particles. See this article by John Baez for more on the nature of Hawking radiation.
The calculation showing that Hawking radiation exists is done by comparing what a quantum field looks like to an observer at infinity in the presence of a horizon. Hawking radiation is seen because observers near and far from the horizon will disagree about what constitiues a vacuum. I can't say any more about this because I don't understand it well enough. If you're interested this paper describes the calculation in layman's terms (for a small subset of extraordinarily well informed laymen).
So the question does a Big Rip pull apart virtual particles is probably meaningless, though the question could presumably be rephrased along the lines of how a Big Rip affects the apparent QFT vacuum. Even then I'm not sure this has an answer. The Hawking calculation is done by extrapolating to infinity and requires an asymptotically flat or effectively flat spacetime. How you'd do the calculation during the approach to the Big Rip I don't know.
If you were an observer in the closing stages of a Big Rip you would see yourself surrounded by a spherical horizon, emitting Hawking radiation, and the horizon would shrink in towards you at an exponentially increasing rate. I'd guess the Hawking radiation would become more intense as the horizon sweeps towards you but i don't know if it would tend to infinity or not.