Exporting a simple model with texture from Blender

Note: I want to avoid modifying the model in the javascript code and do all the model design inside Blender.

Note #2: While this question is long, it is actually a basic problem (title says it all). The below is "walk-through" to the problem.

I am trying export Blender models to threejs.org as DAE model but have problem with models with texture (I have tried JSON and OBJ+MTL format too):

To make things simpler, here are the steps I perform (and fail) to add texture to simple "Startup file" which contains a cube, camera, and light:

Select the cube
In the Materials panel, make sure the default Material for the cube is selected.
In the Texture panel, make sure "Tex" (the default texture for the material) is selected.
For this texture, set Type to "Image or Movie"
In the Image section of panel, open a grass1.jpg (a 512x512 image) as the texture.
In the Mapping section, change the Coordinates to UV.
Export the model as Collada model, checking "Include UV Textures", "Include Material Textures", and "Copy" checkboxes.

You can download blend, dae, and texture file mentioned in these steps.

Then I use the following code to load the DAE model, but I get this error and the cube is not shown:

256 [.WebGLRenderingContext]GL ERROR :GL_INVALID_OPERATION : glDrawElements: attempt to access out of range vertices in attribute 2 WebGL: too many errors, no more errors will be reported to the console for this context.

<script src="js/three.min.js"></script>
<script src="js/OrbitControls.js"></script>
<script src="js/ColladaLoader.js"></script>
<script>
    var scene, camera, renderer;
    init();
    animate();

    function init() {
        scene = new THREE.Scene();
        var WIDTH = window.innerWidth,
            HEIGHT = window.innerHeight;

        renderer = new THREE.WebGLRenderer({antialias:true});
        renderer.setSize(WIDTH, HEIGHT);
        document.body.appendChild(renderer.domElement);

        camera = new THREE.PerspectiveCamera(45, WIDTH / HEIGHT, 0.1, 10000);
        camera.position.set(10,10,10);
        scene.add(camera);

        window.addEventListener('resize', function() {
        var WIDTH = window.innerWidth,
            HEIGHT = window.innerHeight;
            renderer.setSize(WIDTH, HEIGHT);
            camera.aspect = WIDTH / HEIGHT;
            camera.updateProjectionMatrix();
        });  

        var loader = new THREE.ColladaLoader();

        loader.load( 'models/untitled.dae', function(geometry) {
            dae = geometry.scene;
            scene.add(dae);
            var gridXZ = new THREE.GridHelper(100, 10);
            gridXZ.setColors( new THREE.Color(0x8f8f8f), new THREE.Color(0x8f8f8f) );
            gridXZ.position.set(0,0,0);
            scene.add(gridXZ);
        });

        controls = new THREE.OrbitControls(camera, renderer.domElement);
    }

    function animate() {
      requestAnimationFrame(animate);
      renderer.render(scene, camera);
      controls.update();
    }
</script>

And here is the screenshot of Blender after mentioned 7 steps:

Update: Exporting as js file using JSON exporter for Blender doesn't work either and resulted the very same error.

Update 2: Same error after exporting to OBJ+MTL and loading them with OBJMTLLoader.

The problem is you have not set up UV coordinates for your model. By default, each face applies the whole texture, but in blender the UVs are blank when exporting.

You want to specifically set up your UV coordinates. These are coordinates that show how to apply a texture to each face.

Make sure to UV unwrap your model in blender. Go to edit mode (tab), select all faces, press "u", and click "unwrap". Then try to re-export.

Unwrap is just 1 method, there are many. Experiment with different methods in blender to get the results you want (possibly the "reset" option).