Instructions
Turn on the Xbox and allow it to fully start. When prompted, choose to sign in to Xbox Live or allow the console to connect automatically.
Press the "Guide" button located in the middle of your Xbox 360 controller. Hold the right trigger button until the "Settings" blade is selected. Scroll down to select "Profile", then select "Auto Sign-in".
Select the "Disabled" option underneath the "Xbox Live" headline, then press the green "A" key. Restart the Xbox to confirm it no longer automatically signs in to Xbox Live.