Not really. It's barely a whisper, just enough to stay connected.

If the file is really popular, you might get occasional download requests
for parts of it but you can control that by going to the library view, finding it
(in Saved Files), right-clicking on that file and making sure it's not shared.
You can do that to the Saved Files folder too. The odds of anybody actually
downloading the new file are tiny but this will make the odds go down to zero.