It depends on what you use to build the war.
If you use directly the jar command, you should probably create a script that writes the date in the application.properties file and launches the war creation.
If you use ant you can do this in two steps:
- Create a class like this one:
package tests;
import java.io.FileOutputStream;
import java.io.IOException;
import java.util.Date;
import java.util.Properties;
public class GeneratePropertiesFile {
    public static void main(String[] args) throws IOException {
        Properties properties = new Properties();
        properties.setProperty("build.date", new Date().toString());
        properties.store(new FileOutputStream(stagingDir + "/WEB-INF/classes/application.properties"), "");
    }
}
- Write an ant build file like this:
<project name="sample" basedir="." default="buildWar">
    <target name="buildProperties">
        <java fork="true" failonerror="yes" classname="tests.GeneratePropertiesFile"/>
    </target>
    <target name="buildWar" depends="buildProperties">
        <war destfile="myapp.war" webxml="src/metadata/myapp.xml">
            <fileset dir="src/html/myapp"/>
            <fileset dir="src/jsp/myapp"/>
            <lib dir="thirdparty/libs">
                <exclude name="jdbc1.jar"/>
            </lib>
            <classes dir="build/main"/>
        </war>
    </target>
</project>