Concrete Does NOT burn.

More Info: When considering a house fire, concrete structures do not burn like wood and other similar building materials.  In fact, concrete walls have been known to block a fire from spreading into adjacent rooms.

When considering skin contact, concrete CAN burn.

Concrete Structures Are Highly Fire Resistant

Concrete does not immediately disintegrate under the stress from the heat of a normal house fire. Under most conditions, it doesn't begin to become affected by a heat source until it has reached thousands of degrees Fahrenheit, much higher than a typical house fire.  Insulate concrete forms (ICF) also impede the spread of fire by slowing the spread of the flame as well as hindering the transfer of the heat of the fire through the wall.

Concrete Structures Are Not Fireproof

That doesn't mean that a concrete structure is indestructible to fire.  Concrete that has been heated and placed under stress will begin to spall due to the uneven expansion of the various materials that were used in the concrete mixing process.  This breakdown of the concrete is dependent upon many variables including water content, aggregate used in the mix, and rapid changes in temperature can break down the bonds that hold concrete together, causing it to crumble. 

Concrete Spalling Is Highly Variable

The interaction between concrete and heat is highly variable.  For example, newly poured concrete is more susceptible to spalling due to its higher water content.  The aggregate material used in the concrete mixture is another variable that will affect its interaction with heat.  As an example, light-weight concrete, that is concrete that is mixed with a more porous aggregate such as pumice or slag in place of rock, does not conduct heat as efficiently.  The addition of reinforcing materials such as wire mesh and rebar can also affect how the concrete interacts with heat.