The texture on the bunny works, because the rendering of the basic model works with ListSurfacePlot3D, in this case. If the model rendering is not good the application of a texture will not improve the situation.

With ExampleData["Geometry3D"] it seems generally to be better to use RegionPlot3D/DiscretizeGraphics compared to the alternative ListSurfacePlot3D/VertexData.

Examples with the Klein Bottle:

RegionPlot3D@DiscretizeGraphics@ExampleData[{"Geometry3D", "KleinBottle"}]

ListSurfacePlot3D@ExampleData[{"Geometry3D", "KleinBottle"}, "VertexData"]

First get the data from the Wolfram database:

heartModel = AnatomyData[Entity["AnatomicalStructure", "Heart"], "Graphics3D"];
furryTexture = Texture[ExampleData[{"ColorTexture", "CheetahFur"}]];

Then:

RegionPlot3D[
    DiscretizeGraphics@heartModel,
    PlotStyle -> furryTexture
 ]

Dont know exactly what went wrong for you. But it probably has something to do with ListSurfacePlot3D/VertexData versus RegionPlot3D/DiscretizeGraphics. Compare these two examples:

shuttleModelA = ExampleData[{"Geometry3D", "SpaceShuttle"}];
RegionPlot3D@DiscretizeGraphics@shuttleModelA

shuttleModelB = ExampleData[{"Geometry3D", "SpaceShuttle"}, "VertexData"];
ListSurfacePlot3D[shuttleModelB, MaxPlotPoints -> 50]

I originally had those options used, changing the max plot points only changed how intricately the model was mangled.

ListSurfacePlot3D[
Import["C:\\Users\\Ellis\\Documents\\theheartP4.obj", "VertexData"], 
PlotStyle -> Texture[ExampleData[{"ColorTexture", "CheetahFur"}]], 
MaxPlotPoints -> 35, Mesh -> None, PlotRange -> All]

I have also used a few different models in case somehow the vertex data was incorrect prior to projecting a texture onto it.

The heart object that I used was the model from wolfram, but I still had a mangled end result.