Yes, everything will go to hell in a few months' time.
As for the planets flying off into space, from what I understand, the weapon consumes energy from the star, not the mass. Obviously, removing energy would eventually remove mass, causing the orbits to widen, but it would take years, centuries, or even millenia, for this to have much effect. It all depends on how much mass is extinguished. Even if the mass was completely gone, the biggest influences over the first weeks would be lack of sun, not lack of gravitational pull. The planet would leave on a tangent, but not do anything major otherwise. "Flying off into space" for a planet means, well, nothing really, since it's already in space.
Now, taking into account the lack of sunlight, yes, that will have an immediate effect. The most immediate effect would be lack of light on the surface. Yes, that sounds obvious, but it is the most profound. The existing weather systems will keep the temperatures of the planet fairly stable for days or even weeks. As Escoce states the planet would have no more incoming radiation, but it has what is already in its weather systems and rocks and such. And, like others state, there is a huge source of energy seeping heat into the planet, keeping the interior warm. Planets don't radiate much heat to space because there is not much matter in space (it's space, a near vacuum) meaning space can't take the radiated heat. The people on the planet would be quite comfortable in the interior of the planet, since they have heat sources and light sources and no real reason to need the sun anyway.