I am going to assume you have a flame rectification system for proving flame. Make sure all wires to the pilot/spark unit are well secured. The usual problem with this type of flame proving system is the current flow through the pilot flame may become too weak for the control board and it thinks there is no flame, thus shutting everything down. The coincidence of the blower motor starting and the system shutting down may be due to the accompanying voltage drop with the blower starting and that voltage drop causes a decrease in the flow of current through the pilot.
Here's a test to see if I may be right. Set the blower at the thermostat to ON. Then, while the blower is running, call for heat and see if the furnace behaves like it has been... shutting down after a few minutes of operation.