Burning ivory decreases the available supply of ivory in the market (in the diagram below, the supply decreases from S0 to S1). The equilibrium price of ivory increases as a result (from P0 to P1). Higher ivory prices increase the potential profits from poaching, and incentivise more poachers to hunt elephants and existing poachers to hunt more intensively. That hardly seems like a recipe for saving elephants.
A better approach is to target the buyers of ivory, through import bans, fines, imprisonment, etc. If buyers were targeted rather than sellers, that decreases the demand for ivory (from DA to DB in the diagram below). That lowers the equilibrium price of ivory (from PA to PB). The lower prices decrease the potential profits from poaching, and many poachers will find other more attractive activities to engage in (maybe they start poaching rhinos instead).
Is it that easy? Probably not. Under current rules, elephant ivory that originated before the current ban on sales of ivory is allowed to be sold (see here, for example). To make matters worse, as Stephen Gallagher from City University Hong Kong noted in this 2013 article from the South China Morning Post:
Policing the illegal ivory trade has been complicated by the introduction of an exception for ivory originating from countries which claim to have viable elephant populations. In certain circumstances, ivory in its raw state may be exported under licence from Botswana, Namibia, South Africa and Zimbabwe...
Prosecutions may be further complicated by the sale of products in Hong Kong which may be carved from hippopotamus or walrus teeth, both specified species in Appendices I of CITES, or even mammoth ivory, a material obviously not covered by CITES as the species is already extinct.So, a policy of penalising buyers of ivory would have to deal with legal sources of ivory. On top of that, it would need some way of dealing with falsified credentials attached to illegal ivory, a situation which would surely arise. The U.S. has already moved to make domestic ivory trading more difficult by making buyers prove how and when ivory was imported. U.S. moves alone are likely to be insufficient though - the Chinese are the largest buyers of ivory, and enforcement of any penalties against buyers would need to include those in China. The Economist noted last year that:
a ban on ivory sales combined with clever advertising might work. A campaign supported by stars to wean Chinese consumers off shark fin nudged their government into dropping it from state banquets. Overall demand for the traditional delicacy has since fallen by half.So, it may be possible to save the elephants through reduced demand. Ultimately though, the best approach may be to target both buyers and sellers. Reducing both the demand and supply of ivory (as in the diagram below, demand reducing from D0 to D1 , and supply reducing from S0 to S1), reduces the quantity of ivory traded by more than either reducing demand or supply alone would. The effect on price would be ambiguous - it depends on the relative size of the reductions in demand and supply (the diagram shows what would happen if the demand shift were larger in magnitude, with prices falling from P0 to P1).