The lower-back comes into this primarily with respect to your form as you do the sit-up or crunch.
Bent knees, feet on the floor, make your abs work a little harder to lift your upper body.
If you've had back problems, though, crunches, with feet up and crossed, can work just as well, with less stress on the discs.
Bottom line is that if the ab work is hurting your back, either you're doing it wrong or you shouldn't be doing that particular exercise.
Above advice to use a variety of ab exercises is good. Leg or knee raises can be excellent.
I disagree with the advice that squats are all you need for core work.