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I know that I can plot over 3D-regions like balls via

DensityPlot3D[(x^2 + y^2 + z^2)^2, {x, y, z} ∈ Ball[{0, 0, 0}, 1]]

But, not surprisingly, this breaks down for 2D-regions like triangles.

How can I create a 3D plot over surfaces to visualize the functions defined on those surfaces?

I know that there are ways to do this for Balls for example via SliceContourPlot3D (ref Plotting a complicated 3D wave on the surface of a sphere), but I don't think this is possible in this case (and if possible, probably terribly inefficient)

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You don't. DensityPlot3D will reject anything that has RegionDimension not equal to 3, e.g.

In[91]:= RegionDimension@Triangle[{{0, 0, 0}, {1, 0, 0}, {0, 1, 1}}]
(*Out[91]= 2 *)

Instead, you use SliceDensityPlot3D with the 2D region as follows:

SliceDensityPlot3D[(x^2 + y^2 + z^2)^2, 
 Triangle[{{0, 0, 0}, {1, 0, 0}, {0, 1, 1}}], 
 {x, y, z} \[Element] Cuboid[]]

SliceDensityPlot3D of (x^2 + y^2 + z^2)^2 over Triangle[{{0, 0, 0}, {1, 0, 0}, {0, 1, 1}}]

Edit: The symmetry of the function needs to be taken into account, though, when choosing the slice. For example, if your function is spherically symmetric, like (x^2 + y^2 + z^2)^2 is, then "CenterSphere" is a poor choice for slice as the surface will just show numerical noise, e.g.

SliceDensityPlot3D[(x^2 + y^2 + z^2)^2, "CenterSphere", 
 {x, y, z} \[Element] Cuboid[-{1, 1, 1}, {1, 1, 1}]]

SliceDensityPlot3D of (x^2 + y^2 + z^2)^2 over "CenterSphere" showing numerical noise.

But, "CenterCutSphere" works very well here as the numerical noise is not visible relative to the other variations present, e.g.

SliceDensityPlot3D[(x^2 + y^2 + z^2)^2, "CenterCutSphere", 
 {x, y, z} \[Element] Cuboid[-{1, 1, 1}, {1, 1, 1}]]

SliceDensityPlot3D of (x^2 + y^2 + z^2)^2 over "CenterCutSphere" with the default cut-out angle of Pi/2.

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The following works for MeshRegions that represent surfaces.

First, we define a function and a MeshRegion. For simplicity, I use a sphere. A Triangle is not as easy to discretize (i.e. to refine).

f = X \[Function] X[[1]] X[[2]];
R = DiscretizeRegion[Sphere[{0, 0, 0}, 1], MaxCellMeasure -> 0.00001];

Next, we generate a color gradient image with tick lines as texture:

colfun = ColorData["SunsetColors"];
ticks = 20;
tickthickness = 5;
spread = Round[1024/ticks];
n = (spread + tickthickness) (ticks + 1);
a = Developer`ToPackedArray[List @@@ (colfun /@ (Range[0, n]/n))];
Do[
 Do[
  a[[i + j]] = {0., 0., 0.}
  , {j, 1, tickthickness - 1}
  ]
 , {i, 1, n, spread + tickthickness}
];
tex = Image[ConstantArray[a, 15]]

enter image description here

Finally, we evaluate the function on all MeshCoordinates, rescale them and plot the mesh as GraphicsComplex with according VertexNormals (thanks to this post) and with VertexTextureCoordinates according to the (scaled) values of f:

p = MeshCoordinates[R];
vals = Map[f, p];
Graphics3D[{
  Texture[tex], EdgeForm[],
  GraphicsComplex[
   p,
   Polygon[MeshCells[R, 2][[All, 1]]],
   VertexNormals -> Region`Mesh`MeshCellNormals[R, 0],
   VertexTextureCoordinates -> 
    Transpose[{Rescale[vals, {Min[vals], Max[vals]}, {0.001, 0.999}], 
      ConstantArray[0.5, Length[p]]}]
   ]
  },
 Lighting -> "Neutral"
 ]

This is the result of the procedure:

enter image description here

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