Is there a way for me to do sit ups without my back feeling like it's on fire? Or is there another exercise for my abs that doesn't require equiptment?

I use a workout ball to do sit ups on. I make sure my chin is directed toward the celling and never my chest, this requires practice at balance and I feel the burn in my stomach and never my back. I do small lifts to the point it really burns then back down to a flat position on the exercise ball. You might give it a try.

The workout ball is a good idea. I did that also when my back was hurting.

For the lower ab's I just moved my hips up and down but not moving your back at all. If you haven't enough power you might not be able to do that.