If the insurance company take or dispose of the car, they will want the log book off you. When they get it, it will be straight down the salvage auction, and there is a good chance it will be repaired by the buyer and sold on. It is quite common for one buyer to buy two write offs the same make / model, and use the parts from one to repair the other.
If you "buy" the wreck from the insurance company, they wont ask for the log book, but they will inform the DVLA who will request that you submit the car for VIC (vehical identity check) before buying your next tax disc.
