According to the United States Postal Service, it is illegal to place items into a mailbox with the intent of avoiding postage. Flyers, advertisements and other items that are placed in or on a mailbox without paid postage or by someone who is not a Postal employee are considered illegal.
Sunday newspapers are exempted because the Postal Service doesn't operate on Sundays. While the Postal Service recognizes that some people may use mailboxes as a way to drop things off, such as a key for a neighbor, if a mailbox is full, the Postal Service cannot deliver the customer's mail. About.com notes that this regulation remains in effect, even during a Postal Service shutdown when mail is not being delivered.