If you are concerned about the resale value then keep the boxes.
When buying used stock I'm not too worried about the lack of a box - I run the stock, not the boxes. However, if the seller wants "top dollar" pricewise then I want the box. To me the price is relative to the condition of the item & that include the box (or lack of).
BTW - a loft full of empty boxes &/or packaging would act as a good accelerant in the event of a fire.
FWIW I don't believe James May threw the box away - after all we did not see him actually bin it. Maybe he was in Clarkson mode that day.