Keep the boxes for the ref numbers for replacement parts, same as with spare parts bags, keep the tags so I know whats in my buggy and know the prices at my LHS as well as knowing the order number, just makes life a bit quicker at the shop when you know what to order or what to look for.
As for the old pieces, throw them away, there is nothing as irritating as going through your kit bag thinking you have the parts there just to find that its even worse than the piece you want to change, like glow plugs, how many people keep old glow plugs or half stripped screws,nuts and bolts, busted up bearings and old clutch shoes. What about old tyres, just cannt bring myself to throw them away.