I will furnish an answer based on the assumption that the dodecahedron is intended to pass through a hole which is in a plane perpendicular to a fivefold axis. If you were to rotate the dodecahedron to a different orientation (e.g., along a threefold or twofold axis), the shape of the hole would not only be different, its area may actually be _smaller_.
With this in mind, the cross-section's shape is obviously a regular decagon. Some elaborate computation shows that, if the dodecahedron has an edge length of $1$, then the side length of the smallest such decagonal hole is $$\sqrt{\frac{5 + \sqrt{5}}{10}} \approx 0.850651.$$ The area of this hole is $$\sqrt{\frac{5(25+11\sqrt{5})}{8}} \approx 5.56758.$$