According to coinflation.com, the melt value of a current-issue U.S. Nickel is about 6.6 cents. Is it wise for me not to spend any nickels I get in change,  but instead to save them?

Yes. There is a free and open market for older US currency that is worth more than it's face value. See: @http://en.wikipedia.org/wiki/Junk_silver Currency worth significantly more than it's face value gets taken out of circulation rather quickly and traded at a more realistic value. Note that no non penny change you see is older than 1960. This will eventually happen to nickels as well. It may be a while. But with recent drop in precious metals prices, nickels are no longer worth more than 5 cents.

As the other answers point out, mining currency for metals is illegal. I'm not sure whether or not it can be done cost effectively (it might on a large scale). On the other hand, there are people who are stockpiling copper pennies, and even nickels, in the hopes that the government will eventually discontinue them, and they'll no longer be legal currency.   Now, pennies are no longer made of pure copper, but if you collect the ones that are, they contain about 2 cents worth of metal, and that's only likely to go up. Some people have a few hundred pounds of copper pennies sitting in jars, waiting until the government wises up and discontinues them.