Set the line color at the start and at the end.

	// Adds a lineRenderer to this transform and
	// makes the line renderer to fade at the end.
	var c1 : Color = Color.white;
	var c2 : Color = Color(1,1,1,0);
	function Start() {
		var lineRenderer : LineRenderer = gameObject.AddComponent(LineRenderer);
		lineRenderer.material = new Material (Shader.Find("Particles/Additive"));
		lineRenderer.SetColors(c1, c2);
	}

using UnityEngine;
using System.Collections;

public class Example : MonoBehaviour {
    public Color c1 = Color.white;
    public Color c2 = new Color(1, 1, 1, 0);
    void Start() {
        LineRenderer lineRenderer = gameObject.AddComponent<LineRenderer>();
        lineRenderer.material = new Material(Shader.Find("Particles/Additive"));
        lineRenderer.SetColors(c1, c2);
    }
}

import UnityEngine
import System.Collections

public class Example(MonoBehaviour):

	public c1 as Color = Color.white

	public c2 as Color = Color(1, 1, 1, 0)

	def Start() as void:
		lineRenderer as LineRenderer = gameObject.AddComponent[of LineRenderer]()
		lineRenderer.material = Material(Shader.Find('Particles/Additive'))
		lineRenderer.SetColors(c1, c2)