I believe most female audience face-palmed at the prospect of the ugly sonovabitch melting away in the lava.
As always, Hollywood is not the place to learn science.
The buoyancy of an object is dependent on the density of both medium. For example, a piece of wood can float on water because its density is less than that of water. The Dead Sea, the salt lake bordering Jordan, Israel and West Bank, is so dense with salt you'd find floating easier than swimming. And if you push a floating cork into the water, the cork will return to the surface when you release it from your hand.
Viscosity, on the other hand, describes the internal resistance of a liquid to flow. If you drop a metal key into a bowl of soup and a glass of water, you'll probably find the key sinking faster in water while sinking slowly in the soup.
Similarly, if you want to die by sinking into the lava in Mount Doom, you have to be a Terminator or Transformers or you'll find yourself sitting on top of the hot magma. An average human has a density of ~1010kg/m3--slightly more than water, which is why sometimes we float on water and sometimes we don't. But magma has a density of 3100 kg/m3 and a viscosity of 100-1000 Pa*s.
The gas steaming from the magma could reach at least 400 °C. The gas will melt your entire respiratory system within seconds-- a quick death, albeit a painful one.
Back to Gollum's case; he probably won't sink into the lava, but the hot gas should kill him during his fall, way before he could lay his hands on his precious.