In most of the cases the answer is NO, you can't export a .war with an embedded database created with eclipse.
You have to export your database files separately and upload them on the production server. This usually means that you have to install the database server you used while developing and export your DDL and table data.
In your particular case (MySQL with Tomcat) we can say that MySQL is not built for embedded use, unless using an out of date and not quite complete MySQL Embedded Library for Java, which would led you to some major changes in your project DB configuration.
You can altough change your DB server, switching from MySQL to Derby or H2 (thanks to @specializt for this )which have the possibility to embed the server directly in your war .
This is taken directly from an Eclipse Corner Article on Creating Database Web Applications with Eclipse.
BEWARE!!: the linked article is deprecated thus you should not rely on it completely. The quoted part is still sensitive for your case and covers a Derby database deploy.
Deploying the Database
When you are ready to deploy the application to a production Tomcat
application server, you must copy the Derby database data folders to
the application server computer. You then have a couple of options for
accessing the Derby database from your deployed application.
You can install Derby on the application server computer, and run it
in client/server mode. Your application connects to the database using
the same technique as I've demonstrated in this article. The drawback
to this approach is that you have two server processes to manage:
Tomcat and Derby.
You can use Derby in embedded mode. In this scenario, the Derby engine
runs inside the Tomcat process. You don't have to stop and start Derby
separately from Tomcat. For more information, refer to the Derby
website.