Ok I fixed the problem, I just changed the "OnGetButtonDown" in the moving left and right for "OnGetButton", and added two bools to check in which direction the player is moving.

Here is the code, tested and working ;D:

var stillRight : UnityEngine.Texture;
var stillLeft : UnityEngine.Texture;
var movingRight : UnityEngine.Texture;
var movingLeft : UnityEngine.Texture;
var isWalkingLeft : boolean = false;
var isWalkingRight : boolean = false;

function Update () {
    //Moving Left 
    if (Input.GetButton("Left") && !isWalkingRight)
    {
        renderer.material.mainTexture = movingLeft;
        isWalkingLeft = true;    
    } else {
        isWalkingLeft = false;
    }
    //Moving Right
    if (Input.GetButton("Right") && !isWalkingLeft)
    {
        renderer.material.mainTexture = movingRight;
        isWalkingRight = true;
    } else {
        isWalkingRight = false;
    }
    //Still Left
    if (Input.GetButtonUp ("Left"))
        renderer.material.mainTexture = stillLeft;
    //Still Right
    if (Input.GetButtonUp ("Right"))
        renderer.material.mainTexture = stillRight;
}

You put quotation marks in the bottom. Those are strings, not textures. But your variables aren't textures, either. They're materials. So get rid of all of the .mainTexture's, too.

This will help your code break at compile time instead of runtime: http://forum.unity3d.com/threads/71445-How-To-Set-Project-Wide-pragma-Directives-with-JavaScript

Ok so your script could go like this:

var stillRight : UnityEngine.Texture;
var stillLeft : UnityEngine.Texture;
var movingRight : UnityEngine.Texture;
var movingLeft : UnityEngine.Texture;

function Update () {
    if (Input.GetButtonDown ("Right"))
        renderer.material.mainTexture = movingRight;

    if (Input.GetButtonUp ("Right"))
        renderer.material.mainTexture = stillRight;

    if (Input.GetButtonDown ("Left"))
        renderer.material.mainTexture = movingLeft;

    if (Input.GetButtonUp ("Left"))
        renderer.material.mainTexture = stillLeft;
}

What where you doing wrong was:

• you were trying to remplace the player's material main texture with another material, you had to remplace them with another texture so instead of using UnityEngine.Material, you have to use UnityEngine.Texture when declaring your variables.
• you were adding " " in your if functions.

I tested it and it works, I hope this helps you :D