All interactions at the quantum dynamical level are subject to an analysis with Feynman diagrams, and the Casimir effect is not different in this case. Virtual pairs are internal lines in Feynman diagrams.
R.L.Jaffe has a arXiv paper that goes into the detail on the Casimir force and corresponding diagrams.
In discussions of the cosmological constant, the Casimir effect is often invoked as decisive evidence that the zero point energies of quantum fields are "real''. On the contrary, Casimir effects can be formulated and Casimir forces can be computed without reference to zero point energies. They are relativistic, quantum forces between charges and currents. The Casimir force (per unit area) between parallel plates vanishes as "alpha", the fine structure constant, goes to zero, and the standard result, which appears to be independent of "alpha", corresponds to the "alpha"--> infinity limit.
He separates the Casimir effect from the vacuum fluctuations effect.From the conclusions:
I have presented an argument that the experimental confirmat
ion of the Casimir effect does not establish the reality of zero point fluctuations. Casimir forces can be calculated without reference to the vacuum and, like any other dynamical effect in QED, vanish as α→0. The vacuum-to-vacuum graphs (See Fig. 1) that define the zero point energy do not enter the calculation of the Casimir force, which instead only involves graphs with external lines. So the concept of zero point fluctuations is a heuristic and calculational aid in the description of the Casimir effect, but not a necessity.