nope it shouldn't make a connection.. and if you haven't got a live dashboard well wouldn't that suggest you don't have a live account?
if you have a live account, well it doesn't matter to you if you're xbox is on with modchip enabled.....
either way, the only time if can affect you is if you specifically access the xbox live tab on teh main dashboard (which is what you're upgrading your current non-live dashboard to eventually have)....