I've really got to wonder what kind of 7.62x39 magazine would fit, much less feed, 5.56x45 rounds.
But, I would say that it is PERHAPS possible for a 5.56 round to chamber in a 7.62 chamber. The neck and shoulder on the 5.56 round might enter the throat area of the 7.62s chamber, allowing the bolt to close.
For years there have been "HELL, I WAS THERE!" kind of rumors about the Soviets designed their weapons so that they could capture, and fire, our ammo, but not the other way around.
Despite the obvious timeline problems with that argument (the 7.62x39 came out long before the 5.56, or even the .308/7.62x51), as others have noted accuracy is non existent, there's the possibility that the much smaller case will rupture and jam the gun, ad nauseum.