The Fisherman’s Ring, along with a lead seal, is a symbol of the temporal powers of each pope. As such, when the seat of the pontiff is vacant – usually by death, but in this case by renunciation – the ring and the seal can no longer be used for official purposes, such as validating documents. In actual practice, the Fisherman’s Ring, which is large and unwieldy, is rarely used. The tradition comes from a ring that someone would wear and also use with sealing wax on important documents.
The ring is destroyed so that it cannot be used for any other purpose. To auction off such a piece would make profane (that is, secular or not sanctified) something that had been used for a sanctified purpose. Can you imagine what would happen if the new owner started sealing documents or contracts with the Fisherman’s Ring, so that they could be potentially seen as official papal documents? That would make no sense. The ring cannot be sold, as it is far too important a symbol.
In a not-too-similar way, since 1950 the Academy of Motion Picture Arts and Sciences has asked all winners of an Oscar to sign an agreement that states that if the winner or his/her heirs ever choose to sell the statue, they must first offer the Academy the option to buy it back for $1. Pre-1950 Oscars have been auctioned, some for more than six-figure prices. Under the winners agreement, AMPAS preserves the “sanctity” of the award, preventing it from becoming just another commodity.