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It can look dumb, but I always had this question as a kid, what physical principles would prevent this?
The pole would basically be a space elevator. I suspect gravity and inertia would effectively keep you from moving the stick. Even if you could move it, you'd only be able to move it at a speed that would seem like it's stationary. As such, the light would still be faster.
I enjoyed a lot of the discussion in the comments
What about the mass of that stick? Inertial doesn't care for your little silly games.
Neither do the two gravity wells the stick spans. And the earth and moon are moving relative to each other, someone would probably get their head knocked off by that stick. Before it eventually falls to the earth with quite a bit of force because earth's gravity well will win. Then it'll eventually settle into a giant teeter totter, assuming it is rigid enough to survive the impact.
I don't see this mentioned in any of the other comments: the repulsion between atoms that causes the movement to propagate through the stick is actually communicated via photons. So your push really generates the same kind of particles that your light torch is generating, and they travel at the same speed. Except in the stick it is slowed down by repeated absorption and excitation by the electrons of the atoms.
That would not work. Pushing an object is transmitting kinetic energy to it. The object will push back, and energy would not be distributed to the whole object at the same time.
If the object cannot be altered in any way, then the energy would not be transferred to it, and if it has enough plasticity to absorb the kinetic energy, it would be spread in a wave to the tip. A wave that would always be slower than light.
Now stop fooling around and give Ruyi Jingu Bang back to Sun Wukong.