It does not. Why would it? Unless you tell it to.
There is a very valid concern here in case of a Callable for example. If you waited for the result for say 20 seconds and you did not get it, then you are not interested in the result anymore. At that time you should cancel the task at all.
Something like this:
Future<?> future = service.submit(new MyCallable());
try {
future.get(100, TimeUnit.MILLISECONDS);
} catch (Exception e){
e.printStackTrace();
future.cancel(true); //this method will stop the running underlying task
}