Why not? each piece of the lens should focus the light individually on the same focal point - there's nothing magic about it.

You just need to remember that the focal point is generally in the centre of the lens. If you break a chunk off the corner, it will still focus the light to that point - so for example, if you cut the lens into 4 pieces and use one of the pieces, then it will focus the light at the corner which was in the centre of the original lens, not in the centre of the "new" lens

I did this accidentally when my Fresnel credit-card broke and a small piece, about an inch across separated from the main piece. It did focus the light, but not nearly powerful enough to start a fire or anything.

A large Fresnel lens may not have the same optical quality as a smaller one (i.e. the focal point may be larger) in which case the cut piece may or may not be as good as a lens the same size. But it should be easy to check - just mask the large lens with a sheet of paper and see what happens.

They don't work as well as a classic lens for things like photography. I think it was they prism the color just enough to see, but they work well for less demanding (color wise) applications. I wish I could remember exactly, it was a neat show on Discovery.

Frenchy's link above offered a little more info after he posted it and I read it.

Even a classic lens has chromatic abberation, but because the Fresnel has surface breaks its resolution isn't as good. I suspect that's why they aren't used in photography (or astronomy). I don't actually know for sure, just my guess.

I've never used this type of material for that purpose, and I dont know about its optical properties. Applying general priciples though, you should utilize the center of the fresnel sheet. The center should be the "sweet spot" as the fresnel rings converge there. If you cut a small piece from the edges, it may only direct light in one direction, or disperse the light in a scattered pattern. To create heat, the light should be focused from all directions to the lens' focal point.

Experiment yourself or maybe someone else has some good practical experience with this. Good luck.