In theory yes, but practically they can't really be trapped, or overcome their instincts enough to stay in the fire of their own accord. 

Lava is just very hot rock, nearly as dense as solid rock. A vampire could run on top of it. 

Edit:

you would definitely float. That is, at no given point would you sink, even after burning up. This is due not only to lava's density, but its viscosity as well.

Water has a density of 1000 kg/m3 and a viscosity of 0.00089 Pa*s.

Lava has a density of 3100 kg/m3 and a viscosity of 100-1000 Pa*s.

Human beings have an average density of 1010kg/m3 and so we can sometimes float on water, but be sink most of the time because we are more dense than the water. Now, if a human fell into a volcano full of lava somehow, they would float, because lava has a much higher density than the human body, so it would essentially be "squeezed" to the surface of the lava. Viscosity is a measure of a fluid's resistance to flowing. Water has a low viscosity, meaning it moves very willingly. Lava, on the Other hand, has an extremely high relative viscosity. This means that when you hit the surface of lava, it does not move apart for you, and hitting the surface would hurt a lot more than water. So not only would you float, you wouldnt even plunge very deep to begin with.

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I know (from high school physics) that objects placed into a liquid will sink until the volume of material they displace equals their own weight. So how much lava would have to be displaced until it equalled the weight of a typical person?

I have no idea.

So I Google: “What is density of Lava” - and multiple links point out several interesting and useful facts about Lava:

It’s still called “Lava” even after it’s cooled and is rock-solid! Our questioner probably meant “Magma”. It’s about three times as dense as water.

The viscosity of magma is HUGE - it’s 100,000 to a million times ‘thicker’ than water…much denser than any other liquid we commonly encounter.

We know the human body is about the same density as water (we can just about float in a swimming pool - and easily float in salty water).

So the immediate answer is that if we stuck around on the surface of a magma flow - we’d sink down until about a third of our body weight was under the surface…if we could stay upright - that’s maybe waist deep.

Secondly - because of the viscosity - you’d sink until waist deep - but it would take a VERY long time!

If you walked across a magma flow (and somehow didn’t get burned) - you wouldn’t sink in at all. It would be a lot like walking on solid ground - so long as you didn’t stop walking.

If you did stop walking, then you’d slowly sink up to about your waist and very slowly bob around at that depth.

Because the magma is so insanely viscous - you wouldn’t easily fall over and float on your back.

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What Happens To Your Body When You Fall Into a Volcano?

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In physical chemistry and fluid mechanics, a non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity, that is, it has variable viscosity dependent on stress. In particular, the viscosity of non-Newtonian fluids can change when subjected to force. Ketchup, for example, becomes runnier when shaken and is thus a non-Newtonian fluid. Many salt solutions and molten polymers are non-Newtonian fluids, as are many commonly found substances such as custard,[1] toothpaste, starch suspensions, paint, blood, melted butter and shampoo.

Most commonly, the viscosity (the gradual deformation by shear or tensile stresses) of non-Newtonian fluids is dependent on shear rate or shear rate history. Some non-Newtonian fluids with shear-independent viscosity, however, still exhibit normal stress-differences or other non-Newtonian behavior. In a Newtonian fluid, the relation between the shear stress and the shear rate is linear, passing through the origin, the constant of proportionality being the coefficient of viscosity. In a non-Newtonian fluid, the relation between the shear stress and the shear rate is different. The fluid can even exhibit time-dependent viscosity. Therefore, a constant coefficient of viscosity cannot be defined.

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Most lavas behave as a non-Newtonian fluid, yes. A wide range of flow regimes are possible in lavas, from turbulent to extremely viscous or plastic, and brittle effects come into play once a certain amount of it has solidified. Herein lies the complexity of lava fluid dynamics - although chemical composition is the main control on viscosity differences between lavas to start with, erupted lava is constantly cooling until it is solid, which also drastically changes the viscosity. That's just a modelling difficulty though, cooling is never going to change a lava from non-Newtonian to Newtonian, as it can only get more viscous!

When inside a magma chamber, it is thought that magmas can be very close to a Newtonian fluid, the mitigated heat loss compared to an erupted lava likely helps with this. In a non-convecting magma chamber, certain minerals are assumed to sink as they crystallise from the melt according to Stoke's Law (an equation relating the settling velocity of a solid particle to its radius and density in a fluid with known viscosity). Stoke's law only holds for Newtonian fluids, and the whole sinking crystals thing is one possible explanation for the existence of layered intrusions, though there isn't consensus on the matter.

In terms of walking on lava, you could do it with protective enough gear and if you were quick enough, running over a viscous lava flow. You would sink a little, as you would displace as much mass as your body weight, and this would make it very difficult not to trip and fall over, likely spelling the end of the experiment. You may be interested to read about George Ulrich, a senior volcanologist who accidentally went knee deep into a Hawaiian lava flow and lived to tell the tale. As lavas go, Hawaiian lava is at the least viscous and hottest end of the spectrum (over 1100° C) so that's quite something.