Actually, in the books the king of the iron islands doesn’t have the horn, his brother (passed over for the throne) does and arrives in slavers bay as the battle with the maesters is taking place. The show has not even introduced or contemplated that horn. Also, the books don’t make it clear that it is doing anything other than burning the person blowing it.
The horn was introduced in the show... he brought it with him to the kings moot and blew it when he put forth his claim to the throne.
I think it will turn out that the only ones who can blow the horn are those with Valyrian blood in the first place. That’s why it is burning everyone else, they don’t have that “blood of the dragon” fire resistance that Dany seems to have.
Watch the Kingsmoot scene again.