I think the author of this piece has not fully understood the concept of proof-of-work. Hashing algorithms are used precisely because they do not rely on a private key - and thus everyone can verify work that has been claimed to be done.
Passports signing some data would provide a different output for every passport. You could verify that the person doing the hashing indeed is the person who owns the passport if you distributed the public key. But what is the point of this?
Additionally there are other issues on so many different levels with this idea: lack of anonymity, an artificially constrained hash rate (if there's an incentive to increase the hash rate, someone _will_ extract the private key from their passport and mine on dedicated hardware), wearing out the NFC chip through massive use, creative an incentive to steal passports...