I'm not even remotely knowledgeable about this, but I'd assume saltwater would wreck plumbing. The connection in and out would probably be degraded much quicker. Then the water treatment plants would have to deal with dirty salt water, which is probably more difficult.
On top of that all the brine that people produce in their homes would have to be disposed of, and I'm sure many people would just end up flushing it down the drain. So the water treatment plant would have to deal with highly concentrated, contaminated saltwater.
Interestingly, there is one house that I know of with both hot and cold freshwater plumbing as well as hot and cold salt water plumbing: the Breakers mansion, built by the Vanderbilts. I'm sure they spent fortunes maintaining that plumbing and think the tour guide said something to that effect, but everything was a show of wealth there. One room featured platinum wallpaper, because, why not?
Having worked on and plumbed boats, the bigger issue is actually growth and especially mineral deposits. Corrosion is less of an issue since most plumbing is actually plastic at this point. Although any water that goes into an appliance needs to be fresh water, so it would really ONLY be for showering.
Sewage in particular will create hard deposits in plumbing that needs to be dealt with every few years at a minimum.
Frankly, unless you are in a rather extreme environment, like a desert, or a boat where you have to carry or make all your own fresh water, saving a few gallons on showering and washing is pretty inefficient. You could have a far larger impact by changing habits, and ensuring low flow appliances.
On top of that all the brine that people produce in their homes would have to be disposed of, and I'm sure many people would just end up flushing it down the drain. So the water treatment plant would have to deal with highly concentrated, contaminated saltwater.