From Nookipedia, the Animal Crossing wiki
A lost item is a special item in Animal Crossing: New Leaf that randomly appears on the ground outside. Belonging to a random villager in town, the player can ask any villager if they lost an item. If the villager did lose the item, then the player will be rewarded with fruit, clothing, or furniture. If the item does not belong to that villager, they will return it and wish the player luck. It appears in multiple different forms around the town including a book, brown bag, and a mitten. If the item given is a fruit, it will not be a Perfect Fruit. Lost Items cannot be thrown away or sold. There can only be one lost item in the town per day that may be returned to a villager.
If the item is not returned within a day, it will change in the villager's inventory to an item titled 'What was this?' The only option will be to open it. This means that players can choose to keep the lost item and forgo increasing friendship with a neighbor. Also, if the player sells the lost item to Reese, it will cost Bells to sell it.
Not delivering the item on time will lower the friendship between the player and villager, most likely the same amount as denying a request from a villager.