In order to switch off reminders for completed tasks in Notezilla , Go to the sticky note, left click on top left note button and choose "Due Date" and then click on the "Remove" button.
Alternatively press CTRL+R key from within the note and click on the "Remove" button in the Set Due Date window.
Also, please note that Checklist option allows you to tick off the box that is added at the beginning of the line. So that you can track tasks completed. It has nothing to do with reminders.
Learn more about the reminder feature through this short video: