Try setting the credentials on the web service like this:
// -- Windows LiveID Client SDK --
using Microsoft.WindowsLive.Id.Client;
...
private void FunctionWorkingWithListsService()
{
IdentityManager identityManager;
WindowsLiveAuthenticationModule authModule;
Lists listsWebService = new Lists();
listsWebService.Url = listServiceUrl;
listsWebService.UserAgent = listsWebService.UserAgent + ";" +
WindowsLiveAuthenticationModule.WindowsLiveClientHeader;
listsWebService.Credentials = new NetworkCredential();
// -- DO WORK --
}
This is not an answer specific to Windows Phone, but answers the more general question of how to authenticate to a web service using Windows LiveID.
Just make sure the user account has the correct level of permissions on the list to perform whatever actions you are trying to perform.
Microsoft has a full code sample of connecting to an Office Live web service using Windows LiveID here. That's where I pulled the basis of this code from.