Basically our PS3 has tried to eat 2 dvds so far. They still work (with a little encouragement) but the entire edge of the disc looks like someone has run sandpaper around it and little bits of plastic stick out.

What could cause this?

Could dust get in there and rub on it (wood heater is a nightmare for dust)? If so how would we clean it?

It's most likely due to overheating or dust. If it's overheating, it's probably if the vents on your system are blocked. If it's dust, there are shops that can clean or repair whatever's going on.