I have a service that does some work on an HttpServletRequest object, specifically using the request.getParameterMap and request.getParameter to construct an object.
You would generally test these sorts of things in an integration test, which actually connects to a service. To do a unit test, you should test the objects used by your servlet's doGet/doPost methods.
In general you don't want to have much code in your servlet methods, you would want to create a bean class to handle operations and pass your own objects to it and not servlet API objects.