I want to be able to automatically deploy a Java application to a tomcat server.
Current situation:
If you use Maven I will suggest to use Cargo plugin which supports all Tomcat's version and have ability to deploy to local and remote servers.
If you prefer not to go with Maven or the Cargo plugin you can simply invoke the Tomcat manager deploy URL. Under the covers Cargo is basically doing the same thing. Tomcat also provides Ant tasks to wrap the manager commands. Using the Ant tasks would enable you to invoke them in Eclipse without a plugin.
I believe Cargo can do what you want in a general fashion (but I have not tried personally) http://cargo.codehaus.org/Tomcat+6.x
Also Tomcat has an administrative webapp that can accept WAR's by upload, but I would look at Cargo first.