According to NOAA, the ocean was originally not very salty but became saltier over time as rivers eroded the land and delivered the dissolved minerals to the ocean. At the same time, salts crystallize out of the water and are deposited on the ocean floor. This input and output are now more or less balanced so the ocean is not getting saltier. Apparently, this salt cycle involves about 4 billion tons of new dissolved salts being added to the ocean each year and about the same amount being deposited from the water to the ocean bottom.

So, why aren't rivers salty? Apparently, it is because rivers carry only a small amount of salt and are kept fresh by constant rainfall, whereas the ocean has been accumulating salt for the last 4 billion years.

Lakes that don't drain to the ocean, like the Dead Sea, can get salty over time, just like the ocean.

To some extent, these compounds will inevitably mix together. During the early stages of earth (hadean period), there was a time when it was raining all the time, which meant that all of the minerals on the surface were exposed to water. Naturally, some of those were water soluble, which changed the composition of the growing oceans at the time. Some minerals also underwent various other reactions, which caused them to crumble (weathering) which exposed even more reactive surface. In some cases, you ended up with cracks that allowed the rain water to penetrate deeper into to the crust and find its way to larger deposits of water soluble minerals, such as NaCl. The initial exposure to water only kickstarted the process, but later rain and rivers continued to deliver even more salt to the oceans, resulting in the current salinity over the course of billions of years.

In order to prevent the initial dissolution of salts, you would need to have a planet without oxygen in any form, so that there would not be any water. If your planet has oxygen and water, but no chlorine, you would still get various other salts such as sulfates, which would make the oceans salty. Either way, it would be a very exotic combination of elements, and might never actually happen.

If you’re ok with the initial dissolution of salts during the hadean era, but wish to prevent any later dissolution of salts, you could do that by evaporating all the water, just like Venus and Mars did. However, then you won’t have any oceans either, so that’s not ideal.

Another way would be to make the planet as cool as the moons of Jupiter and Saturn, so that there would be hardly any liquid weather. This way, the midly salty oceans produced in the hadean period would be covered with a sheet of ice, preventing any further weathering and dissolution. Also, a Water World (remember that movie) should produce a similar result, since rain and rivers aren’t in contact the rock surface. However, the salt from the hadean period would still be there, so this isn’t ideal either.

The dead sea mechanism is also an interesting alternative. Just replicate that mechanism at a massive scale, and you have relatively fresh water oceans and massive dead seas that just accumulate all of the salt from other bodies of water. Those surface salt deposits would need to be close to the equator so that the sun can evaporate all of the water that flows into them. Those deposits would also need to be lower than the rest of the terrain, and they would need to be connected to the surrounding oceans via rivers, which is a tall order IMO.

Over the course of billions of years, some of those salt deposits might get pushed into the fresh water oceans, which would mess up the whole thing. I think this setup is not stable for billions of years, but it could be possible for a certain period anyway. Maybe this could be a good place for a scifi story. Imagine a planet with massive fresh water oceans and several saturated salt pools near the equator.