Generally, any situation where a charge is intended to reduce demand / disadvantage the payer, rather than to maximise profit / benefit the payee.
At the moment in these sorts of cases usually a payee just keeps the money; the advantage would be in reducing perverse incentives.
If spammers kept opening HN accounts, HN could say "burn $5 to open a new account" to reduce demand. That would avoid HN having an incentive to ban people for profit.
Or the state could say "speeding fine? burn $200" to discourage people from speeding. That would avoid the state having an incentive to increase crime to generate fine revenue.
Or a college could say "want your exam paper re-graded? burn $50" to discourage requests just for the sake of it. That would avoid the college having an incentive to grade people badly.
To resolve a disputed transaction on an auction site, the site could require the buyer to destroy the goods and the seller to burn the money, so there would be no profit from filing false disputes.
Hrm, the ability to do this is actually rather intriguing.
I wonder how feasible it would be to have a service that provides proof-of-burn for real-world money? Effectively, allow people to pay the company money, company then provides burner with a receipt referencing this proof-of-burn. The burner can then provide that receipt to the party that wishes the burn, and the party can verify with this service that the money was indeed burnt (and this verification then marks the burn as having been "used", so to speak). The company could do something like take a small fee from the burnt money (to actually pay for the service), and donate the rest to charity (and maybe even provide the donation receipt to the burner, so they can treat it as a charitable gift for tax purposes).
The big obstacle here is how to ensure that the money is irrevocably burnt, but that's a solvable problem (check or wire transfer, perhaps, or even waiting long enough after a credit card payment to prevent chargeback, although I think that's a rather long time).
The downside here is that you need a middleman to run the whole operation, and irrevocably paying them takes a bit of effort and time.
The upside is the money isn't actually destroyed, it's instead given to charity.
At the moment in these sorts of cases usually a payee just keeps the money; the advantage would be in reducing perverse incentives.
If spammers kept opening HN accounts, HN could say "burn $5 to open a new account" to reduce demand. That would avoid HN having an incentive to ban people for profit.
Or the state could say "speeding fine? burn $200" to discourage people from speeding. That would avoid the state having an incentive to increase crime to generate fine revenue.
Or a college could say "want your exam paper re-graded? burn $50" to discourage requests just for the sake of it. That would avoid the college having an incentive to grade people badly.
To resolve a disputed transaction on an auction site, the site could require the buyer to destroy the goods and the seller to burn the money, so there would be no profit from filing false disputes.