Is there any possibility for my Java (EE) application to get notified by the Team Foundation Server when a Build or Test Run is completed?
Depending on the TFS server version you can use the Service Hooks option to configure a web service to be called when specific events happen. This featrue was introduced with TFS 2015.
If you're on an older version you can configure an alert and instead of configuring it to use an email endpoint, point it to a SOAP webservice. This technology is considered deprecated, but it still works.
I'm unsure whether Test Runs are explicitly included in these alert options.