It has nothing to do with whether or not the water itself is salty. One theory is that some mineral content in the water combines with sodium ions coming from sweat on your hand - and since salt can help clean wounds, it's suggesting that your body needs hydration. Drinking more water may help get rid of the taste in some cases - but for persistent cases, doctors recommend drinking bottled unsalted water instead of one that might have little trace minerals which could continue to muddy up taste receptors.
Taste receptors are found around our mouths in small bumps called papillae. Different types of papillae are sensitive to different tastes - sensing sweet, salty or sour molecules when they come into contact with them.