Climate on your globe will be forever altered, though it's difficult to fully describe how as we really aren't sure what currently drives the process. You risk messing around with thermohaline circulation (also known as the great oceanic heat conveyor belt). One part we do know is this system is dependant on salt water concentrations to run. Water warms in the tropics and moves along the surface of the ocean towards land. In the atlantic, this current keeps England relatively warm and in the pacific it brings wet and warmer weather to the pacific northwest. The water then cools, sinks into the ocean, and returns towards the tropics in a giant loop.
It's actually thought that metling ice could adjust the oceans salt concentrations enough to disrupt this cycle, let alone all salt being removed from it...and the end result is a bit of a 'day after tomorrow' style story, without the silly storms. England, Scotland, and Ireland all enter a freezing cycle where they take on the climate closer to what a land that far north should see (average around -10C colder as a minimum). We would also see the tropics water warm further and surface temperatures would start to reach record highs relatively quickly. Hurricanes use the surface heat of the ocean to fuel itself (a large hurrican actually drags cold water up from the depths and kills its own fuel source)...with a larger fuel source it stands to reason these systems could become significantly stronger and we could see some monster storms.
We would get frozen icecaps pretty quickly as well...saltwater affects the temperature at which water freezes, lowering it significantly and allowing for cold ocean water not to freeze right away. Funny enough, this would actually have quite the warming effect on the globe as water mass transitions to ice and releases the energy involved with it's freezing. Speculative, but you might see a 2-5 degree temperature increase on world wide averages pretty quickly too. This would be temporary as I suspect the ice formed would begin reflecting large amounts of sunlight reaching earth and start a cooling trend.
The tropical areas of your globe will become much more humid (especially near these oceans) and I suspect you'll see a significantly larger amount of rain. If nothing makes up for the thermohaline circulation, you'll also see the equator gain a tremendous amount of heat and keep it. So warm wet tropics with tons of rain and potentially the most fierce hurricanes we've witnessed.
There's no possible way of resolving the impact to the ecosystem either...very few species are capable of crossing salt water into fresh water and this salt precipitation event would likely become a mass extinction event that would rival the meteor impact that ended the dinosaurs as far as number extinct species is concerned. We're not talking a few salt water fish either, we're talking every last living creature in the ocean, including many that we're not even aware of yet. I'm not sure if there is a single fresh water Cephalopod for example. Good chance you'd cause a massive plankton die off with this as well.
I'm actually thinking you could post an answer to the question 'how to I end life on earth' with 'remove salt from the oceans' and be somewhat correct.