The Kinetic Energy of an object moving at non-relativistic speeds is calculated as Ke = ½mv² which implies that speed is v = (2Ke/m)^0.5. To affect the speed I can either increase my body's energy output or lower my useless body weight. So if I lower my weight by one kilogram, but keep my energy output the same how much faster can I run?
So we see v_final = v_initial x (m / (m - 1))^0.5. So one kilogram weight reduction shows me I can increase my velocity by (m / (m - 1))^0.5 kph or in seconds 60/((m / (m - 1))^0.5). Finally, plotting in my current weight of 120 kg. I get that a 1 kg weight loss could make my pace increase by ¼ of a second, so 4 kg will drop my pace by a second. Since I feel I am about 40 kg overweight, that means I could drop my pace at this energy output to 10 seconds a km or 50 seconds on a 5k race! That's pretty significant. This means that I should try to lose weight consistently in healthy amounts over my training.
But because I showed today that I can increase my running pace with proper milestones and have a big result on the final time, I would consider weight-loss as a positive bonus in running, but more of a tuning "trick" to improve my time.