Im using ODP.net for 11g with no connectivity problems but I turned off the oracle listener and was still able to connect and pull data.
Does the Oracle ODP use the Oracle Listener?
All remote connections need to go through a listener. Once connected, the listener may not be required again (but it isn't something you'd turn off).
Local connections don't require a listener, but it is good practice to go through the listener as it replicates the remote functionality.
How did you turn off the listener ? And is the database local or remote ?
It is unlikely (not impossible, but VERY unlikely) that the shortcut in the Windows menu is turning off the remote listener. I suspect you have a local Oracle DBMS install (possibly without an actual database, just the software) and you turned off the local listener. The remote listener will still be up and running. You don't need a local listener for a database connection.