Does increasing energy mean there is an external force? So, in one of the online lectures that I was watching, Walter Lewin did an experiment to prove the law of conservation of momentum. 
There were two cars, attached to each other by a spring (The cars have equal masses) Initially, they are at rest. The spring has some pent-up potential energy. Then he burns the spring, and now the cars gain velocity in opposite directions. Since momentum has to be conserved, in the absence of any external force, the ratio of velocities of the cars should be 1. All well. The experimental data matches too. 
The experiment can be found here: Lect 15 — Momentum, Conservation of Momentum, Center of Mass.
It starts from the 29th minute. 
I am confused because when he applies heat to the spring to move the cars from rest, then isn't he supplying energy to the system? So, if energy is being supplied to the system, according to the work-energy theorem, net work done by the force is also increasing right? Doesn't this, therefore, imply that either force or displacement increases? So, shouldn't momentum not be conserved in this case?
 A: Heat is energy that is not available to do work. Therefore, heating the spring does not do work on the system. When you wake up on a cold morning and step into the hot shower do you get pushed in some direction? (If you do then you might need to lower your water pressure or hire a paranormal expert to check out your house).
You could argue that the molecules in the spring have increased kinetic energy due to net work done on them, sure. But this energy transfer will not be able to do work on the macroscopic scale. Heat is just random energy transfer that tends to flow from hot to cold.  
The first law of thermodynamics, which is just energy conservation, states that the change in internal energy $\Delta U$ of a system is equal to the sum of the work done on that system and the heat that enters the system:
$$\Delta U=W+Q$$
In this case $W=0$, so the internal energy increases due to the heat. Energy is still conserved.
A: He doesn't burn the spring, he burns the string! The string holds the spring together and when he burns it, the spring extends and releases its energy. The string cannot push the cars and has no potential energy, because it isn't elastic like the spring.
