Are you able to go to your map service from the web browser using the same URL that you provide in ArcGISTiledMapServiceLayer? Please also try to wire up to Initialized event and check for failure using the following code.
Layer l = sender as Layer;
if (l.IsInitialized || l.InitializationFailure != null)
{
//check for failure.
}
Have you looked at this blog post? http://blogs.esri.com/Dev/blogs/silverlightwpf/archive/2009/08/24/Troubleshooting-blank-layers.aspx