Here comes the science:
They state that the metal is 2800 degrees. If we assume this is in F, then we can reasonably assume that the metal is iron because 2800F is the melting point of iron. Aside from that, iron is a very common metal for casting. So, we're going to assume that we're working with iron.
According to P.92 of Thermodynamics: An Engineering Approach, 5th edition, we have the following formula for heat conduction:
Qdot = kA(delta T / delta x)
We have two volumes separated by a thermally conductive material. Qdot is Q with a dot over it - as in rate of heat transfer (J/s, W). k is the thermal conductivity. A is the area of the contact surface. delta T is the difference in temperatures of the volumes. delta x is the thickness of the separating material.
According to Table 2-3, human skin has a thermal conductivity of 0.37 W/m*K. Human body temperature is 98.6F (310.15K). The metal is at 2800F (1810.9K). According to Wikipedia,
human skin is 1-2mm thick. I think finger skin is on the thin side, so we'll go with 1mm (0.001m). I also think that 20cm^2 (0.02m^2) sounds like a good contact area.
Qdot = 0.37(0.02)[(1810.9-310.15)/0.001] = 11000W = 11J/ms
That means that our hand will be able to absorb 11J of heat per millisecond. So, what do we consider to be too hot for our fingers? I'll go with 120F (322.0K). That means we need an increase of 11.85K.
My book doesn't have the specific heat of human, so I went with "beef round". From Table A-3, beef round has a specific heat of 3.08kJ/kg*K (3.08J/g*K). I also couldn't find the density of either substance, so I'm just going with 1g/cm^3 (density of water). So, if we assume 1cm thick flesh over our 20cm^2 contact area, we have 20cm^3 of flesh. This gives us 20g.
To find how much heat we need to raise the temperature:
delta Q = m * c * delta T
delta Q = 20 * 3.08 * 11.85 = 729J = ~730J
So, we need 730J to burn our hand. With a transfer rate of 11J/ms, we can keep our hand in the metal for 66ms.
Addendum:
I forgot to mention why working with iron was important. Actually, it probably isn't as important as the fact that it has a higher thermal conductivity than skin. The thermal conductivity of iron is 80.2 W/m*K. As long as the hot substance has a thermal conductivity higher than that of skin, this analysis will hold. If the thermal conductivity is lower, however, it will be less willing to give its heat up for your hand and you'll have to use the conductivity of the substance. Such substances, listed in Table 2-3, are wood, rubber, and air.