Warning: This AsyncTask class should be static or leaks might occur

Non-static inner classes holds a reference to the containing class. When you declare AsyncTask as an inner class, it might live longer than the containing Activity class. This is because of the implicit reference to the containing class. This will prevent the activity from being garbage collected, hence the memory leak.

To solve your problem, either use static nested class instead of anonymous, local, and inner class or use top-level class.

How to use a static inner AsyncTask class

To prevent leaks, you can make the inner class static. The problem with that, though, is that you no longer have access to the Activity's UI views or member variables. You can pass in a reference to the Context but then you run the same risk of a memory leak. (Android can't garbage collect the Activity after it closes if the AsyncTask class has a strong reference to it.) The solution is to make a weak reference to the Activity (or whatever Context you need).

public class MyActivity extends AppCompatActivity {

    int mSomeMemberVariable = 123;

    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_main);

        // start the AsyncTask, passing the Activity context
        // in to a custom constructor 
        new MyTask(this).execute();
    }

    private static class MyTask extends AsyncTask<Void, Void, String> {

        private WeakReference<MyActivity> activityReference;

        // only retain a weak reference to the activity 
        MyTask(MyActivity context) {
            activityReference = new WeakReference<>(context);
        }

        @Override
        protected String doInBackground(Void... params) {

            // do some long running task...

            return "task finished";
        }

        @Override
        protected void onPostExecute(String result) {

            // get a reference to the activity if it is still there
            MyActivity activity = activityReference.get();
            if (activity == null || activity.isFinishing()) return;

            // modify the activity's UI
            TextView textView = activity.findViewById(R.id.textview);
            textView.setText(result);

            // access Activity member variables
            activity.mSomeMemberVariable = 321;
        }
    }
}

Notes

• As far as I know, this type of memory leak danger has always been true, but I only started seeing the warning in Android Studio 3.0. A lot of the main AsyncTask tutorials out there still don't deal with it (see here, here, here, and here).
• You would also follow a similar procedure if your AsyncTask were a top-level class. A static inner class is basically the same as a top-level class in Java.

• If you don't need the Activity itself but still want the Context (for example, to display a Toast), you can pass in a reference to the app context. In this case the AsyncTask constructor would look like this:

  private WeakReference<Application> appReference;
  
  MyTask(Application context) {
      appReference = new WeakReference<>(context);
  }
  

• There are some arguments out there for ignoring this warning and just using the non-static class. After all, the AsyncTask is intended to be very short lived (a couple seconds at the longest), and it will release its reference to the Activity when it finishes anyway. See this and this.
• Excellent article: How to Leak a Context: Handlers & Inner Classes

Kotlin

In Kotlin just don't include the inner keyword for the inner class. This makes it static by default.

class MyActivity : AppCompatActivity() {

    internal var mSomeMemberVariable = 123

    override fun onCreate(savedInstanceState: Bundle?) {
        super.onCreate(savedInstanceState)
        setContentView(R.layout.activity_main)

        // start the AsyncTask, passing the Activity context
        // in to a custom constructor
        MyTask(this).execute()
    }

    private class MyTask
    internal constructor(context: MyActivity) : AsyncTask<Void, Void, String>() {

        private val activityReference: WeakReference<MyActivity> = WeakReference(context)

        override fun doInBackground(vararg params: Void): String {

            // do some long running task...

            return "task finished"
        }

        override fun onPostExecute(result: String) {

            // get a reference to the activity if it is still there
            val activity = activityReference.get()
            if (activity == null || activity.isFinishing) return

            // modify the activity's UI
            val textView = activity.findViewById(R.id.textview)
            textView.setText(result)

            // access Activity member variables
            activity.mSomeMemberVariable = 321
        }
    }
}