ClearAll[regF]
 regF = PolyhedronData["Dodecahedron", "RegionFunction"];

 regF[x, y, z] // TeXForm

>$\scriptsize\sqrt{10 \left(5+\sqrt{5}\right)} (2 x+z)+5 \left(2+\sqrt{5}\right)\geq 0\land \sqrt{10 \left(5+\sqrt{5}\right)} (2 x+z)\leq 5 \left(2+\sqrt{5}\right)\land
 \sqrt{50-10 \sqrt{5}} x+\sqrt{10 \left(5+\sqrt{5}\right)} z\leq 5 \left(\left(1+\sqrt{5}\right) y+\sqrt{5}+2\right)\land \sqrt{2 \left(5+\sqrt{5}\right)} z\leq
 2+\sqrt{5}\land \sqrt{50-10 \sqrt{5}} x+5 \left(1+\sqrt{5}\right) y+\sqrt{10 \left(5+\sqrt{5}\right)} z\leq 5 \left(2+\sqrt{5}\right)\land 2 \sqrt{5 \left(5+2
 \sqrt{5}\right)} x+10 y\leq \sqrt{10 \left(5+\sqrt{5}\right)} z+5 \left(2+\sqrt{5}\right)\land 2 \sqrt{5 \left(5+2 \sqrt{5}\right)} x\leq 10 y+\sqrt{10
 \left(5+\sqrt{5}\right)} z+5 \left(2+\sqrt{5}\right)\land 5 \left(1+\sqrt{5}\right) y\leq \sqrt{50-10 \sqrt{5}} x+\sqrt{10 \left(5+\sqrt{5}\right)} z+5
 \left(2+\sqrt{5}\right)\land z+\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}}\geq 0\land \sqrt{50-10 \sqrt{5}} x+5 \left(1+\sqrt{5}\right) y+\sqrt{10
 \left(5+\sqrt{5}\right)} z+5 \left(2+\sqrt{5}\right)\geq 0\land \sqrt{10 \left(5+\sqrt{5}\right)} z\leq 2 \sqrt{5 \left(5+2 \sqrt{5}\right)} x+5 \left(2
 y+\sqrt{5}+2\right)\land 10 y+\sqrt{10 \left(5+\sqrt{5}\right)} z\leq 2 \sqrt{5 \left(5+2 \sqrt{5}\right)} x+5 \left(2+\sqrt{5}\right)$

 ClearAll[rotate, rotatedRegF]

 rotate[t_, axis : ("xaxis" | "yaxis" | "zaxis") : "xaxis"] := RotationTransform[t, 
 axis /. Thread[{"xaxis", "yaxis", "zaxis"} -> IdentityMatrix[3]]] @* List;

 rotatedRegF[rf_, t_, axis : ("xaxis" | "yaxis" | "zaxis") : "xaxis"][x_, y_, z_] := 
 rf @@ rotate[t, axis][x, y, z]

 rotatedRegF[regF, t][x, y, z] // TeXForm

>$\scriptsize\sqrt{10 \left(5+\sqrt{5}\right)} (y \sin (t)+z \cos (t)+2 x)+5 \left(2+\sqrt{5}\right)\geq 0\land \sqrt{10 \left(5+\sqrt{5}\right)} (y \sin (t)+z \cos (t)+2
 x)\leq 5 \left(2+\sqrt{5}\right)\land \sqrt{10 \left(5+\sqrt{5}\right)} (y \sin (t)+z \cos (t))+\sqrt{50-10 \sqrt{5}} x\leq 5 \left(\left(1+\sqrt{5}\right) (y
 \cos (t)-z \sin (t))+\sqrt{5}+2\right)\land \sqrt{2 \left(5+\sqrt{5}\right)} (y \sin (t)+z \cos (t))\leq 2+\sqrt{5}\land \sqrt{10 \left(5+\sqrt{5}\right)} (y
 \sin (t)+z \cos (t))+5 \left(1+\sqrt{5}\right) (y \cos (t)-z \sin (t))+\sqrt{50-10 \sqrt{5}} x\leq 5 \left(2+\sqrt{5}\right)\land 10 (y \cos (t)-z \sin (t))+2
 \sqrt{5 \left(5+2 \sqrt{5}\right)} x\leq \sqrt{10 \left(5+\sqrt{5}\right)} (y \sin (t)+z \cos (t))+5 \left(2+\sqrt{5}\right)\land 2 \sqrt{5 \left(5+2
 \sqrt{5}\right)} x\leq \sqrt{10 \left(5+\sqrt{5}\right)} (y \sin (t)+z \cos (t))+10 (y \cos (t)-z \sin (t))+5 \left(2+\sqrt{5}\right)\land 5
 \left(1+\sqrt{5}\right) (y \cos (t)-z \sin (t))\leq \sqrt{10 \left(5+\sqrt{5}\right)} (y \sin (t)+z \cos (t))+\sqrt{50-10 \sqrt{5}} x+5
 \left(2+\sqrt{5}\right)\land y \sin (t)+z \cos (t)+\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}}\geq 0\land \sqrt{10 \left(5+\sqrt{5}\right)} (y \sin (t)+z \cos
 (t))+5 \left(1+\sqrt{5}\right) (y \cos (t)-z \sin (t))+\sqrt{50-10 \sqrt{5}} x+5 \left(2+\sqrt{5}\right)\geq 0\land \sqrt{10 \left(5+\sqrt{5}\right)} (y \sin
 (t)+z \cos (t))\leq 5 \left(2 (y \cos (t)-z \sin (t))+\sqrt{5}+2\right)+2 \sqrt{5 \left(5+2 \sqrt{5}\right)} x\land \sqrt{10 \left(5+\sqrt{5}\right)} (y \sin
 (t)+z \cos (t))+10 (y \cos (t)-z \sin (t))\leq 2 \sqrt{5 \left(5+2 \sqrt{5}\right)} x+5 \left(2+\sqrt{5}\right)$

 rfunctions = {"regF", "rotatedRegF[regF,Pi/2]", "rotatedRegF[regF,Pi/2, \"yaxis\"]", 
 "rotatedRegF[regF,Pi/2, \"zaxis\"]"};

 Grid[Partition[Labeled[RegionPlot3D[ToExpression[#][x, y, z],
 {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, 
 PlotPoints -> 40, ImageSize -> 300, 
 Method -> {"ShrinkWrap" -> True}, Boxed -> False, 
 AxesOrigin -> {0, 0, 0}], #, Top] & /@ rfunctions, 2], 
 Dividers -> All]

[![enter image description here][1]][1]

**Update:** For interactive display, rendering the polyhedron once and using `GeometricTransformation` with desired transformation is faster:

 ir = PolyhedronData["Dodecahedron", "ImplicitRegion"];
 rp = RegionPlot3D[ir, PlotPoints -> 70];

 {xaxis, yaxis, zaxis} = IdentityMatrix[3];

 Panel @ Manipulate[Row[Panel[
 Graphics3D[{{EdgeForm[], FaceForm[], 
 Cuboid[{-3, -3, -3}, {3, 3, 3}], Red, PointSize[Large], 
 Point[ctr]}, EdgeForm[None], FaceForm[Opacity[.5]], 
 GeometricTransformation[rp[[1]], RotationTransform[t, #]]}, 
 Axes -> True, PlotRange -> {{-3, 3}, {-3, 3}, {-3, 3}}, 
 AxesOrigin -> {0, 0, 0}, Boxed -> False, ImageSize -> 300, 
 PlotLabel -> #]] & /@ {xaxis, yaxis, zaxis}], 
 {{t, 0}, -Pi, Pi}, Paneled -> False]

[![enter image description here][2]][2]

Anchor rotation at, say, `vv = {2, 1, 1}`:

 vv = {2, 1, 1};
 centroid = {0.0000305046, -0.0000984894, -0.0000394806};

 Manipulate[Row[Panel[
 Graphics3D[{{EdgeForm[], FaceForm[], 
 Cuboid[{-3, -3, -3}, {3, 3, 3}], Red, PointSize[Large], Point[vv]},
 Dashed, Thick, Opacity[1, Blue], 
 GeometricTransformation[Line[{centroid, vv}], RotationTransform[t, #, vv]], 
 EdgeForm[None], FaceForm[Opacity[.5]], 
 GeometricTransformation[rp[[1]], RotationTransform[t, #, vv]]},
 Axes -> True, 
 PlotRange -> {{-6, 6}, {-6, 6}, {-6, 6}}, 
 AxesOrigin -> {0, 0, 0}, Boxed -> False, ImageSize -> 300, 
 PlotLabel -> #]] & /@ {xaxis, yaxis, zaxis}], 
 {{t, 0}, -Pi, Pi}, Paneled -> False]

[![enter image description here][3]][3]


 [1]: https://i.sstatic.net/z3rBV.png
 [2]: https://i.sstatic.net/lx8GV.gif
 [3]: https://i.sstatic.net/4P48n.gif