Under almost all normal composting conditions No.
The waste and compost is damp (wet). The bacteria creating the heat in the heap die off at 70-75C. The maximum temperature most heaps will achieve is 75C. Most ‘light’ organic chemicals will only ‘self-ignite’ around the 120C mark. The HOTBIN is designed to work HOT (up to 75C), the inside compost rarely over dries. Water vapour and carbon dioxide (both flame retardants) are constantly flowing to the valve. The combination of damp waste, steam and carbon dioxide reduce the chance of combustion. The sun will not create overheating inside the HOTBIN – the insulation works both ways it keeps bacterial heat in but it also keeps the sun's heat out.
Occasionally the newspapers print a report of a fire in compost bin or heap. We have looked up the science and experts reports. I stress no-one at HOTBIN is a combustion expert, but our reading of the reports is that it is near impossible to ignite a small domestic compost heap. It can happen with very large dry heap (above 2m*2m heap is signalled as risk size). Time and again the problem was not traced to a 'normal compost heap' (ie damp, as composting requires 60% water). The papers describe a poorly managed pile of wood or dry plant matter. The reports also indicate that fires in dry waste and/or compost sites are most often linked to the inadvertent addition of hot ashes, cigarettes and other materials that were already alight.
It is known that large haystacks can and do self-combust in very rare circumstances. The damp inner decays creating heat. This heat does not escape fast enough (materials are self insulating) and airflow restricted. If this heat persists for long enough it can lead to ‘self oxidation heating’ and push the temperature up to 120C and hence self ignite. Very rare but possible.
The science indicates is is technically possible, but the situations for it to arise are highly improbable. There is a good (if rather detailed) expert summary of spontaneous combustion of organic matter at: href="http://www.jgpress.com/BCArticles/2000/01004.html"