Let's look at reason why Gold cannot go to $0. I can think of at LEAST 2 reasons:

1) Gold can make very good and reliable electronic contacts.  If gold is cheap then it will be used instead of tin, copper, and other metal that corrode in its environment and make computer and other electronics unreliable
2) Gold will not rust.  I can build a house with roof plated with 1mm worth of gold if its $1/oz (still not $0)
and it will not leak for centuries.   Gold is extremely malleable and unlike copper it won't harden due to wind and mechanical vibration (and it would develop micro fracture which is why copper roof can leak)

Bitcoin can go to $0 in following cases:

1) All coins were made unusable, for example run out of 0's on merkle root so no new transactions can be had due to extremely fast computer/miner that secure the bitcoin network before a hard fork to address these issues,
2) network protocol which bitcoin runs on achieves 100% ban / packet blocked globally.