@John has answered to the second question: "Is there a natural process which removes salt from the sea at a significant rate?" This is an answer to the first one: "Is there a natural limit to this process, or the will the sea keep getting saltier forever?"
Yes, there is a limit, sea water cannot keep getting saltier forever.
Sea water is a solution of water and salt, i.e., the salt is dissolved in the water. But solubility, the ability of a material to dissolve in another, has a limit called saturation. Once this limit is reached, the material cannot dissolve anymore and precipitate.
Saturation of salt in water is 357 g L$^{-1}$ (at 20 °C, it depends on temperature), about 10 times the current salinity of seawater. If you keep adding more salt to such water, it will not longer be able to dissolve but will instead precipitate at the bottom of the sea.