Re: How to mark a reminder as completed from locked screen?
If there is a notification on the lock screen for a specific task, swipe left on it and there should be the option to mark as completed.
Alternately you could set Reminders to show up in your Today screen (swipe down from top). If your tasks have a due date set, they'll show there. From the lock screen you could swipe down to open the Today screen and check them off. But in my testing just now it seemed like they only show up in Today if they have a due date of Today.
Otherwise, if they're just regular tasks that you've put into Reminders, then yeah, you'll have to open the app. Even the 3D Touch doesn't give you the option to mark something as completed.