Does placing a metal spoon in an open bottle of sparkling wine (etc.. .) help keep its fizz?
As a beer brewer, I'm pretty concerned with fizz ;)  Since the below may be a little tl;dr, the short answer to your question is, "I don't think so."

This is actually the first time I've heard of the metal spoon "trick", so I can't directly comment on that, but I'll share a little of what I know about carbonation.

Carbonation is carbon dioxide (CO2) that is dissolved in the liquid (beer, champagne, soda).  The fizz is that CO2 coming out of solution and escaping into the air.

Keeping it cold helps keep the fizz, as CO2 dissolves more readily into a cold liquid.  And conversely, comes out of a warm liquid more readily.  Ever pour a glass of warm soda or champagne straight after opening?  It'll foam over everywhere.

The only way to not lose all that CO2 is to seal the container.  Even then, the CO2 will come out of solution and pressurize the headspace (space in the top of the container).  That's why you get a hiss when you first open a bottle.

Again, temperature comes into play.  In a warm bottle, more of the CO2 comes out of solution and pressurizes the headspace.  Chill that same bottle down, and the CO2 can dissolve back into the liquid, giving you a sparkling beverage again.

Now, back to the spoon.  I would think that putting a spoon in the carbonation liquid would produce nucleation points, causing the CO2 to come out of solution faster.  This is why many beer glass manufacturers are coming out with laser-etched designs on the bottom of their glasses - the effervescence caused by the nucleation points helps bring out the hop aromas.