I have a production server, which doesn't have a JSP engine, but can run Java servlets. Is it possible to copy the compiled JSP pages (which are servlets) to this machine and run them without a JSP engine?

Even better what you can do is use jspc which will comile your JSPs into servlets. The current JSP/servlet engines are starting to support this, and I know it is included with JRun and Tomcat. The current versions also allow you to generate a web.xml file as well so that all your file URLs stay the same, i.e., the JSP at /example/test.jsp will become a servlet called JSPExampleTest (for example), and then in web.xml it will map the /example/test.jsp request to the JSPExampleTest servlet.