Thank you for the information. I think the network do matters. I have seen some limitations when you load the map without network.
However, I tested your scenario with Airplane mode on with the following code and it works.
mMapView.addLayer(new ArcGISLocalTiledLayer(Environment.getExternalStorageDirectory().getPath()+"/ArcGIS/SanFrancisco.tpk"),0);
and activity_main.xml should be:
<!-- MapView -->
<com.esri.android.map.MapView
android:id="@+id/map"
android:layout_width="fill_parent"
android:layout_height="fill_parent">
</com.esri.android.map.MapView>
Is there anything different with your code?