What is the role of non renewable resource in nations development?

The most important thing about non renewable resources for developing countries is ease of use and cheap availability.
To use the energy in a fossil fuel, often times all that is required is fire. Coal, gas, and oil all produce high amounts of energy when burned. Developing countries often lack the infrastructure and technology to adequately use or develop advanced technologies like solar cells and hydro-power, but setting up a coal engine is relatively easy.
To access coal and oil, all you need to know is where the field is and how deep you need to go to reach it. Strip mining and oil drilling easily accessed fields is an easy way to harvest massive amounts of energy.
In addition, fossil fuels can be transported and sold much easier than nearly any other form of energy. As such, they often make up a significant portion in the developing country's GDP.
Eventually, developing countries will reach a point where technology allows them to utilize advanced energy sources in addition to non renewable energy, as happened in the US and Europe.