Understanding the problem
Think of your wiring like a pipe full of water, and each device like a pump. Each pump draws a specific amount of water, but not necessarily the same amount as any of the other pumps, and the water in the pipe can only be supplied at a fixed rate.
When the light turns on, it "pumps" a small amount of water from the supply pipe. This amount is far less than the pipes supply rate, so there is no problem.
When the furnace comes on, it "pumps" a huge amount of water (for a short time). This amount is close to, or greater than the amount of water in the supply pipe. This means there is less water in the pipe for the light, so the light cannot function properly.
This is an over simplified, and arguably inaccurate analogy.
Solving the problem
First off, Having the light on the same circuit as the furnace is not correct. This should be fixed as soon as possible, and will likely solve the problem you are seeing.
This could be an indication that one of the components of the furnace is reaching the end of its life, and needs to be serviced. High current draws in an appliance are commonly caused by motors. Motors will draw more current while starting than they do when they are running, this is known as inrush current and can quite a bit higher than the devices full-load current. This large current draw during start-up can be increased if additional force is needed to get the motor spinning, for example if the bearings are going bad. You'll want to inspect the furnace, and pay close attention to any motors. Make sure they are properly greased, and working normally.
If this has always been an issue all along, the furnace is probably operating normally and there is nothing to worry about. If this has only become an issue recently, you may want to inspect your furnace (or have a professional inspect it).
I'd try bending the hot contact in the socket out a little bit or a different bulb. If the problem persists I'd be inspecting the connections inside the junction box.
Try a different bulb (swap in a known working bulb, and swap this one out to a known working can light).
Try bending the contact out (the pin inside the socket).
Inspect the connections (remove the three screws holding the can light housing in. Pull it out of the ceiling and now you can access the junction box).
Rub the threads of new bulbs in your hair (to lubricate them) before you insert them.
Best Answer
I am pretty sure what you have is classic case of voltage drop. Ballasts for vapor lights and some cheap light sensors for incandescent par bulbs will be tricked or drop output, so light goes out and has to restart when voltage is restored. The reason for the sudden drop is really the concern. Since the heat pump employs a fairly large motor, it draws a lot of current at start up. This is called inductive kick. It takes a few seconds for the coils of the motor to saturate and start he armature spinning in the magnetic field. Since the coils are a dead short until saturation is achieved, they draw large current. E = I/R, ohms law explains this. So what is the solution? You may need to have the heat pump motor checked out or add a motor starting capacitor across the windings. I cannot tell you what size capacitor you will need as it can vary widely depending on motor size. This is a very common problem in residential services that are close to current capacity with an unbalanced motor load. Even though you may have a 100 or 200 amp electrical service, it can be overtaxed temporarily by an unbalanced motor start up