It should be pretty easy to implement this. There are a couple features in LCDS that should make setting this up fairly straightforwad. LCDS has the notion of a bootstrap service that is started when the server starts up. LCDS also has the ability to do server push; ie push messages from the server to a destination that a client application is subscribed to. You could write a bootstrap service that tails your app server log file and then pushes any changes to a messaging destination that the Flash client is subscribed to.
There is some LCDS sample code you can look at that will show you how to both setup a bootstrap service and do server push. In the WEB-INF/src directory of the
lcds-samples webapp take a look at the flex.samples.feed.Feed class. This class uses the MessageBroker.routeMessageToService API to push messages to a destination. Also look at the flex.samples.runtimeconfig.EmployeeRuntimeDataDestination class. This is an example of a bootstrap service that is created and started at server startup.
For the log file tailing, it looks like there are a number of libraries written in Java to do this. This thread mentions some of them.
Hope that helps.