The asynctask will keep running. The asynctask might die as the Android OS tries to reclaim memory when it's running low. If you don't care that the async task might not finish, then maybe this doesn't matter, but it's bad practice.
Also, maybe when the async task dies depending on how you're storing/handling the result, you risk trying to edit an Activity/objects in that activity that are no longer there, resulting in nullpointerexceptions and other bad things. The proper way of doing this would be to start a service from your activity and start an asynctask from the service. You should follow one of the design patterns outlined in this talk given at Google IO 2010:
Android REST Client Design Patterns
That talk interesting but is hard to follow and gives no code, so here's a good tutorial that I used to learn this stuff:
Downloading Data with Services
Also, android will manage a threadpool of your asynch tasks for you (i believe it will start killing them when you have more than 5 in your process), you don't have to kill them. check out the answer here:
How AsyncTasks die