Most salts are ionic, formed by strong base and acid, and are almost indestructible, so I guess nothing will happen
On the other hand, acid MAY dissolve part some insoluble salt (like carbonates)
However, what happen mostly depend on the acid that make the salt. If it is weak (like H2CO3), the salt are dissolve by the stronger acid (carbonates make bubbles when dropped into HCl, for example).
Salt from strong acid doesn't react, even when put into an another strong acid (like BaSO4 doesn't dissolve in water)