ContourPlot using ElementMeshSurfacePlot3D

You could use SliceContourPlot3D. Using the definitions of mr and uif from the page linked in the OP (and defined in user21's answer),

SliceContourPlot3D[
 uif[x, y, z], {mr}, {x, -8, 8}, {y, -5, 5}, {z, -2, 5}, 
 BoxRatios -> Automatic, Boxed -> False, Axes -> False, 
 Contours -> 20, ColorFunction -> "TemperatureMap"]

ElementMeshSurfacePlot3D does not have an option to plot contours in Version 10. It may get one in the future. One thing you can do is use the function imsFindContour form this old, outdated package. You'll find the function in the file UnstructuredPlot.m

Then you can do something like this:

mr = BoundaryDiscretizeGraphics[
   ExampleData[{"Geometry3D", "SpaceShuttle"}]];
uif = NDSolveValue[{Inactive[Laplacian][u[x, y, z], {x, y, z}] == 1, 
    DirichletCondition[u[x, y, z] == 1, z <= -1.3]}, 
   u, {x, y, z} \[Element] mr];

Get["Imtek`UnstructuredPlot`"] // Quiet
Needs["NDSolve`FEM`"]

mesh = uif["ElementMesh"];
Show[
 ElementMeshSurfacePlot3D[uif, Boxed -> False, 
  ViewPoint -> {0, -4, 2}],
 Graphics3D[
  Polygon /@ 
     imsFindContour[mesh["Coordinates"], 
       Join @@ ElementIncidents[mesh["BoundaryElements"]], 
       uif["ValuesOnGrid"], #][[All, -1]] & /@ 
   Range[Sequence @@ MinMax[uif["ValuesOnGrid"]], 2], Boxed -> False]
 ]

Not perfect but maybe a starting point.