How to remedy an over-brined turkey?
Unfortunately, there's no easy way to get salt out of meat (or other types of food) once it's there.  If the turkey weren't cooked yet, it's possible to try to put it in pure water (rather than brine), and gradually some the salt will come back out into the water.  After sitting for a while, replacing the water with more fresh water can speed up the process.  (This is the process sometimes used for desalting pickles that are fermented in very salty brine, but it only can do so much.)

In reality, though, that's not going to help much with moving salt out and could take days, during which your turkey will go bad.  I'm assuming since you tasted it that you've also already cooked your turkey.  So what now?

Unfortunately, I think the only realistic thing to do with leftovers here is to use them to flavor other dishes that are not naturally salty.  For example, you could make a gravy without salt and add the turkey meat to it to simmer, then the salted meat can provide some salt to the gravy for open-faced sandwiches or something.  Or make turkey soup using unsalted broth/stock, but add the turkey meat, which will release some of its salt into the liquid while the soup simmers.

As it's difficult to remove salt, really what you're trying to do is dilute its flavor by adding the turkey meat to other unsalted (or low-salt) things.